# Energy transfer mechanisms in adverse pressure gradient turbulent boundary
layers

Taygun R. Gungor 1,2  Yvan Maciel2  Ayse G. Gungor1<EMAIL_ADDRESS>1
Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469
Maslak, Istanbul, Turkey 2Department of Mechanical Engineering, Laval
University, Quebec City, QC, G1V 0A6 Canada

###### Abstract

The energy transfer mechanisms and structures playing a role in these
mechanisms in adverse-pressure-gradient (APG) turbulent boundary layers (TBLs)
with small and large velocity defects are investigated. We examine the wall-
normal and spectral distributions of energy, production and pressure-strain in
APG TBLs and compare these distributions with those in canonical flows. It is
found that the spectral distributions of production and pressure-strain are
not profoundly affected by an increase of the velocity defect, although the
energy spectra drastically change in the inner layer of the large defect APG
TBL. In the latter, the signature of the inner layer streaks is absent from
the energy spectra. However, the production and pressure-strain spectra
suggest that the near-wall cycle or another energy transfer mechanism with
similar spectral features still exist in large defect TBLs. In the outer
layer, energetic, production and pressure-strain structures appear to change
from wall-attached to wall-detached structures with increasing velocity
defect. Despite this, the 2D spectral distributions have similar shapes and
wavelength aspect ratios of the peaks in all these flows. These observations
suggest that outer layer energy transfer mechanisms may be the same in wall-
bounded flows, no matter if dynamically relevant structures are attached or
detached to the wall. Therefore, the conclusion is that the mechanisms
responsible for turbulence production and inter-component energy transfer may
remain the same within each layer in all these flows. It is the intensity of
these mechanisms within one layer that changes with velocity defect, because
of the local mean shear variation.

###### keywords:

turbulent boundary layer, adverse pressure gradient

##  1 Introduction

When a turbulent boundary layer is subjected to a sufficiently strong or
prolonged adverse pressure gradient, its mean momentum defect increases. This
growing defect progressively changes the nature of the flow, and hence APG
TBLs become different from canonical wall-bounded flows such as zero-pressure-
gradient (ZPG) TBLs or channel flows.

Indeed, the change in the mean velocity profile leads to a different
distribution of the mean shear in the wall-normal direction, which in turn
affects the turbulent energy distribution in the boundary layer. With less
mean shear, the turbulent activity in the inner layer decreases (Skåre &
Krogstad, 1994; Elsberry et al., 2000), and the inner maximum of Reynolds
stresses vanishes in the case of large velocity defect TBLs (Maciel et al.,
2006b; Gungor et al., 2016; Maciel et al., 2018). The outer layer turbulence,
on the other hand, becomes dominant when the velocity defect is important, and
a turbulence production peak emerges in the outer layer (Skåre & Krogstad,
1994; Gungor et al., 2016).

In the inner layer, the impact of the pressure gradient on the coherent
structures also depends on the extent of the mean velocity defect. The
spanwise and streamwise sizes of the most energetic $\langle
u^{2}\rangle$-carrying structures do not change significantly when the defect
is small (Lee, 2017; Harun et al., 2013). Like in canonical flows, the inner
peak of the energy spectra of $\langle u^{2}\rangle$ is at inner-scaled
spanwise wavelength ($\lambda_{z}^{+}$) of approximately $120$ (Lee, 2017;
Tanarro et al., 2020) and inner-scaled streamwise wavelength
($\lambda_{x}^{+}$) of approximately $1000$ (Harun et al., 2013; Sanmiguel
Vila et al., 2020). Furthermore, the shape of the $\langle u^{2}\rangle$
spectra is resemblant in ZPG and small defect APG TBLs (Harun et al., 2013;
Tanarro et al., 2020). When the defect is large, spatial organization and
spectral features of the energetic structures change in the inner layer. The
near-wall streaks are weakened (Skote & Henningson, 2002; Lee & Sung, 2009).
They also become more irregular, disorganized, less streaky (Lee & Sung, 2009;
Maciel et al., 2017a), and even vanish at separation (Rahgozar & Maciel,
2012). Moreover, the inner peak in the $\langle u^{2}\rangle$ spectra, which
is connected to the streaks, vanishes in the large-defect case (Lee, 2017;
Kitsios et al., 2017). Apart from the changes in the streaks, Maciel et al.
(2017a, b) reported that sweeps and ejections are weaker in the inner layer
than in the outer layer in large-defect TBLs, and their number is less than in
ZPG TBLs.

The elevated outer layer turbulence activity in APG TBLs was reported numerous
times regardless of the velocity defect (Harun et al., 2013; Lee, 2017;
Tanarro et al., 2020). The $\langle u^{2}\rangle$ spectra show that the
energetic outer layer structures are longer and wider than their counterparts
in the inner layer in APG TBLs (Harun et al., 2013; Kitsios et al., 2017). The
outer peak of the spectra is at $\lambda_{x}/\delta\approx 3$ (Harun et al.,
2013; Sanmiguel Vila et al., 2020) and $\lambda_{z}/\delta\approx 1$ (Bobke et
al., 2017; Lee, 2017), where $\delta$ is the boundary layer thickness.
Elevated outer layer activity is also reported for high Reynolds number
canonical flows (Hutchins & Marusic, 2007b; Marusic et al., 2010a) with an
outer peak in the $\langle u^{2}\rangle$ spectra due to elongated, meandering
structures (Marusic et al., 2010a), which are called superstructures in
external flows (Hutchins & Marusic, 2007a) and very large-scale motions
(VLSMs) in internal flows (Kim & Adrian, 1999). The streamwise wavelength of
the outer peak in the energy spectra of $\langle u^{2}\rangle$, which is
associated with these structures, is in the order of $6\delta$ in boundary
layers (Smits et al., 2011). There is also another type of structures, which
are called large scale motions (LSMs), associated with smaller wavelengths of
$\lambda_{x}/\delta\approx 2$-$3$ (Smits et al., 2011). The energetic outer
layer structures in APG TBLs are more similar to LSMs than
superstructures/VLSMs in terms of dimensions. However, Sanmiguel Vila et al.
(2020) investigated the effects of APG and high Reynolds number for small-
defect APG TBLs and found two outer peaks (one similar to the ZPG
superstructures’ one, and the typical APG one) in the $\langle u^{2}\rangle$
spectra at sufficiently high Reynolds number. Regarding the magnitude of the
typical APG outer peak, several studies have reported its increase with
increasing velocity defect (Lee, 2017; Sanmiguel Vila et al., 2020) when the
levels are normalized with friction-viscous scales. However, such a conclusion
could be an artefact of using the friction-viscous scales because they are not
proper scales for energy levels for APG TBLs (Gungor et al., 2016), especially
for large defect ones (Maciel et al., 2018).

The turbulence regeneration mechanisms, or in other words, self-sustaining
mechanisms, have been extensively studied in canonical flows. However, studies
concerning APG TBLs are rather scarce. One of the mechanisms that has been
suggested for APG TBLs is the instability of streaks (Marquillie et al.,
2011), which was already considered as a regeneration mechanism for canonical
wall-bounded flows (Hall & Horseman, 1991). Another mechanism proposed for the
outer region of large defect APG TBLs is an inflectional instability
associated with the outermost inflection point of the mean velocity profile,
which is inviscidly unstable. Elsberry et al. (2000) suggested that such an
instability affects the flow field in their large defect APG TBL. The location
of the inflection point was close to that of the maxima of the Reynolds normal
stresses and turbulent kinetic energy production. The presence of an
inflection point at the same wall-normal location as the peak of the Reynolds
stresses was reported by other researchers as well (Gungor et al., 2016;
Kitsios et al., 2017). Furthermore, Schatzman & Thomas (2017) suggested that
an embedded shear layer, which is centered around the aforementioned
inflection point in the outer layer, exists in APG TBLs. They proposed scaling
parameters based on this idea of an embedded shear layer and obtained self-
similar mean velocity and Reynolds stress profiles in their large-defect APG
TBL, as well as in other large-defect cases. Balantrapu et al. (2021) also
obtained self-similar profiles with the same scaling in their highly
decelerated axisymmetric turbulent boundary layer. However, Maciel et al.
(2017b) noted that they could not find any roller-like structures that are the
sign of a Kelvin-Helmholtz type instability. In addition, it is important to
note that moderate-defect APG TBLs have an outer maximum of the Reynolds
stresses without the presence of an inviscidly unstable inflection point in
the mean velocity profile (Maciel et al., 2018). It is still not known whether
an inflection point instability exists, and furthermore if it is directly
responsible for turbulent regeneration in large defect APG TBLs, but it is
possible that the presence of Reynolds stress peaks and an inflection point in
the outer layer is simply correlated without any causality (Balantrapu et al.,
2021).

From a different perspective that does not necessarily involve an inflectional
instability, several researchers have reported that APG TBLs with large
velocity defect might behave like free shear flows due to the change in the
mean velocity (Gungor et al., 2016; Kitsios et al., 2017). Gungor et al.
(2020) demonstrated that the Reynolds shear stress carrying structures in
large defect APG TBLs and homogeneous shear turbulence flow, which has no
inflection point, have similar shapes and are mostly dependent on the local
mean shear. These studies suggest that self-sustaining mechanisms might be
similar in the outer layer of large defect APG TBLs and free shear flows and
highlight the causal role played by mean shear.

The task of identifying the self-sustaining mechanisms present in APG TBLs is
a formidable one. A first step along that route is to better understand the
energy transfers directly resulting from the self-sustaining processes, namely
turbulence production and energy transfer between Reynolds stress components
(the pressure-strain term in the Reynolds stress transport equations).
Examining the wall-normal distributions of production and pressure-strain does
not suffice for that purpose. A more complete picture can be obtained by also
investigating the spectral distributions, among scales and position, of these
energy transfers. The idea of studying the spectra of the terms in the
Reynolds stress transport equations was introduced first by Lumley (1964), and
it has gained attention recently. The spectra of the turbulent kinetic energy
or Reynolds stress transport equations have been utilized to examine spatial
transport, energy cascade, or scale separation in channel flows (Mizuno, 2016;
Cho et al., 2018; Lee & Moser, 2019), Couette flows (Kawata & Alfredsson,
2018) or ZPG TBLs (Chan et al., 2021). As mentioned above, in the present work
we focus on the spectral features of production and pressure-strain with the
aim of better understanding the self-sustaining mechanisms present in APG
TBLs.

More precisely, the goal of this study is to examine the spectral
distributions of energy, production and pressure-strain in inner and outer
layers of APG TBLs and compare them with the ones in canonical wall-bounded
flows to better understand the mechanisms involved. We consider both small and
large velocity defect cases to analyze the effect of velocity defect on these
energy transfer mechanisms.

##  2 Databases

We utilized three types of flows in this paper: APG TBLs, ZPG TBLs, and
channel flows. A new DNS database is generated for APG TBLs. This new database
is an extension of the database that was introduced by Gungor et al. (2017)
and analyzed by Maciel et al. (2018), which was referred to as DNS2017. The
main difference between the new database and DNS2017 is that the new database
has a wider spanwise width than the previous one. Other than that, both
databases are almost identical with minor differences in the boundary
conditions. Since they are almost identical, we will only use here the most
recent one. The details of the databases employed in this paper are given in
the following sections.

$(a)$$(b)$$(c)$$(d)$$(e)$ Figure 1: Streamwise development of the main
parameters of the APG TBL. The suction/blowing boundary condition ($V_{top}$)
and edge velocity ($U_{e}$) $(a)$. Pressure gradient parameters $\beta_{ZS}$
and $\beta_{RC}$ $(b)$. Shape factor, $H$, and skin friction coefficient,
$C_{f}$ $(c)$. $\delta$, $\delta^{*}$ and $\theta$ normalized with their value
at inlet $(d)$. Reynolds numbers $\Rey_{\theta}$ and $\Rey_{\tau}$. The white
zone shows the useful range. The open circles denote the three streamwise
positions that are used in the paper.

###  2.1 The current APG TBL

The current database is a DNS database of a non-equilibrium APG TBL with a
Reynolds number based on momentum thickness ($\Rey_{\theta}$) reaching up to
$8000$. The code used to perform the DNS solves the three-dimensional
incompressible Navier-Stokes equations. It provides the time evolution of
three-dimensional velocity and pressure fields for a given flow configuration.
The DNS code employs a hybrid MPI/OpenMP approach for parallelization. Further
details on the code can be found in Simens et al. (2009) and Borrell et al.
(2013).

The DNS computational setup consists of two simulation domains, the auxiliary
and main domains, running concurrently as described in Sillero et al. (2013)
and Gungor et al. (2016). The auxiliary ZPG TBL DNS, with a coarser
resolution, is intended to provide the realistic turbulent inflow data for the
main APG TBL DNS. The goal of using two domains is to provide inflow
conditions for the main DNS with lower computational cost. Regarding the other
boundary conditions, the bottom surface is a flat plate with a no-slip
boundary condition. The side boundary conditions are periodic. The far-field
boundary condition is adjusted in the form of suction and blowing to apply
adverse/favorable pressure gradients. The outflow condition is a Neumann
boundary condition. The computational domain of the main APG TBL simulation, a
rectangular volume with a streamwise, wall-normal and spanwise length of
$(L_{x}$, $L_{y}$, $L_{z}$)/$\delta_{av}$ = $25.3$, $4.8$, $7.1$, is
discretized with $N_{x}$, $N_{y}$, $N_{z}$ = $4609\times 736\times 1920$ grid
points. Here the average boundary layer thickness ($\delta_{av}$) is
calculated within the useful range (or so-called domain of interest), which is
shown as a white zone in figure 1.

Figure 1 shows the spatial development of main parameters of the APG TBL. As
stated before, the pressure gradient in the APG TBL is generated by imposing a
suction/blowing boundary condition for the wall-normal component of the
velocity at the top boundary ($V_{top}$). Figure 1a shows the spatial
development of $V_{top}$ along with the edge velocity, $U_{e}$. The definition
of $U_{e}$ is not straightforward in flows with inviscid velocity that varies
in the wall-normal direction, such as the current flow. Although it is not the
most generally applicable definition (see for instance Griffin et al. (2021)),
we utilize the maximum streamwise component of the mean velocity in the wall-
normal direction as $U_{e}$ for this study to be consistent with most
literature (Gungor et al., 2016; Maciel et al., 2018) and because it works for
the flow cases studied here.

At the beginning of the domain, the suction velocity sharply increases within
a short distance to impose the desired pressure gradient distribution.
Downstream of this region, the suction velocity is adjusted to obtain a
regularly increasing shape factor $H$ (figure 1c). The change in $V_{top}$
near the end of the computational domain is to increase the numerical
stability of the simulation by accelerating the flow so that the Neumann
boundary condition still holds. The suction boundary condition decelerates the
flow, which is seen from the development of the edge velocity ($U_{e}$) in
figure 1a.

Figure 1b shows two pressure gradient parameters that characterize the effect
of the pressure gradient on the outer layer: Rotta-Clauser pressure gradient
parameter ($\beta_{RC}$) and the pressure gradient parameter based on the
Zagarola-Smits velocity ($U_{ZS}$). They are defined in equation 1.

$\beta_{RC}=\frac{\delta^{*}}{\rho u_{\tau}^{2}}\frac{dP_{e}}{dx},\hskip
14.22636pt\beta_{ZS}=\frac{\delta}{\rho U_{ZS}^{2}}\frac{dP_{e}}{dx}$ (1)

Here, $\delta^{*}$ is the displacement thickness, $\rho$ is the density,
$u_{\tau}$ is the friction velocity and $P_{e}$ is the pressure at the edge of
boundary layer. $\beta_{RC}$, which is the traditionally used pressure
gradient parameter, progressively increases until $\beta_{RC}$ is above $350$
and then sharply decreases. However, as Maciel et al. (2018) demonstrated, it
is not a valid pressure gradient parameter for TBLs with large velocity
defect. $\beta_{ZS}$, which correctly represents the local impact of the
pressure gradient on the boundary layer regardless of velocity defect (Maciel
et al., 2018), increases near the flow entrance up to 0.35 and then decreases
to approximately 0.05 and remains mostly the same in the last part of the
domain. The rapid increase of $\beta_{ZS}$ at the beginning of the domain
causes the increase in momentum loss in the boundary layer ($H$ increases,
figure 1c). The subsequent decrease of $\beta_{ZS}$ over most of the domain
should theoretically lead to momentum gain in the boundary layer (H
decreases). However, because the boundary layer responds with a delay to the
pressure force evolution, the momentum gain only occurs at the end of the
domain.

The effect of the pressure gradient on the flow is also seen in the
development of the skin friction coefficient, $C_{f}$, as shown in figure 1c.
$C_{f}$ steadily decreases until around $x/\delta_{av}\approx 20$ and becomes
very close to zero. The behavior of $C_{f}$ and $H$ reflects the strong non-
equilibrium nature of the current APG TBL flow.

Figure 1d shows the spatial development of $\delta$, $\delta^{*}$, and
momentum ($\theta$) thicknesses. They all increase until close to the end of
the domain. Figure 1e shows the distributions of the most commonly used
Reynolds numbers for TBLs, $\Rey_{\theta}$ and $\Rey_{\tau}$. $\Rey_{\theta}$
increases approximately from 2000 to 8000. Reynods number based on $u_{\tau}$,
$\Rey_{\tau}$ develops irregularly in the streamwise direction. However, it is
not valid because $u_{\tau}$ is not a valid scale for AGP TBLs with large
velocity defect. The irregularity of $\Rey_{\tau}$ stems from utilizing
$u_{\tau}$ as a velocity scale.

###  2.2 Existing Databases

Because we focus on spectral analysis in the current paper, we have two
criteria for selecting existing databases of ZPG TBLs and channel flows:
sufficiently high Reynolds number and availability of spectral distributions
of Reynolds stresses, production and pressure-strain. The DNS database of Lee
& Moser (2015) with a $\Rey_{\tau}$ of 2000 is employed for channel flows.
This database is chosen because some one- and two-dimensional spectra are
available as a function of streamwise and spanwise wavenumbers ($k_{x}$,
$k_{z}$) and $y$. The DNS database of Sillero et al. (2013) with a
$\Rey_{\theta}$ of $6500$ and the experimental database of Baidya et al.
(2021) with a $\Rey_{\theta}$ of $6191$ are chosen for ZPG TBLs. We chose
these two databases for ZPG TBLs because the Reynolds numbers are similar and
the 1D spectral distributions of Reynolds stress and production are available
as a function of $k_{z}$ and $y$ for the database of Sillero et al. (2013) and
$k_{x}$ and $y$ for the database of Baidya et al. (2021). They complement each
other for the spectral analysis without introducing any significant Reynolds
number effect.

Name | Type | Database | $\Rey_{\theta}$ | $\Rey_{\tau}$ | H | $C_{f}\times 10^{-3}$ | $\beta_{ZS}$ | $\beta_{RC}$ | Color & Symbols
---|---|---|---|---|---|---|---|---|---
APG1 | APG TBL | DNS | $3005$ | $646$ | 1.65 | 2.1719 | 0.30 | 4.5 |
APG2 | APG TBL | DNS | $4395$ | $593$ | 2.00 | 1.0674 | 0.17 | 13.6 |
APG3 | APG TBL | DNS | $5787$ | $460$ | 2.63 | 0.3599 | 0.08 | 40.1 |
ZPGa | ZPG TBL | DNS | $6500$ | $1990$ | 1.35 | 2.7063 | $\approx 0$ | $\approx 0$ | $\bm{-}\mathrel{\mkern-14.0mu}\bm{-}$
ZPGb | ZPG TBL | Exp. | $6191$ | $2493$ | 1.35 | 2.7602 | $\approx 0$ | $\approx 0$ | $\circ$
CH | Channel | DNS | NA | $1995$ | NA | 3.3605 | NA | NA |
Table 1: The information about the databases used in the paper.

###  2.3 Flow Description

We aim to investigate a wide range of velocity defect cases in this study. For
the APG TBL, three streamwise positions from the APG TBL are employed. These
streamwise positions, as shown in figure 1, represent three velocity defect
cases, and their corresponding shape factors are $1.65$, $2.00$ and $2.63$.
The shape factor is $1.35$ for both ZPG TBL cases since they are at similar
Reynolds numbers. The velocity defect is smaller in the ZPG TBL than the small
defect case of the APG TBL. As for the channel flow, the velocity defect,
which is with respect to the centerline velocity, is smaller than those of the
ZPG TBLs as it is well-known. These five flow cases, details of which are
provided in Table 1, cover a wide range of velocity defect situations from a
channel flow to a TBL with a large velocity defect.

Regarding the Reynolds number of the cases, the ZPG TBL and the channel flow
cases have similar $\Rey_{\tau}$ and are selected at higher $\Rey_{\tau}$ and
$\Rey_{\theta}$ than the APG TBL cases. We could have chosen different
databases for ZPG TBLs or channel flows with a lower Reynolds number instead.
However, it is better to analyze canonical flows at high Reynolds number
because of the elevated outer layer activity in such flows (Hutchins &
Marusic, 2007a). In this manner, we can compare the outer peaks of the
spectral distributions of canonical flows and APG TBLs.

Throughout the paper, the streamwise, wall-normal and spanwise directions are
referred to as $x$, $y$, and $z$ or $1$, $2$, and $3$ for index notation. The
corresponding instantaneous velocity components are $\bar{u}$, $\bar{v}$, and
$\bar{w}$. The brackets, $\langle.\rangle$, denote ensemble averaging.
Furthermore upper cases and lower cases denote the mean value and the
fluctuations, respectively. Thus, $\langle\bar{u}_{i}\rangle=U_{i}$ and
$\bar{u}_{i}=U_{i}+u_{i}$. The upper index plus means the friction-viscous
scales, with the friction velocity $u_{\tau}$ as the velocity scale and
$\nu/u_{\tau}$ as the length scale.

$(a)$

$(b)$

$(c)$

$(d)$

Figure 2: The mean velocity profiles of all databases normalized with outer
scales (a) and friction-viscous units (b). The mean shear profiles of the DNS
databases normalized with outer scales (c) and friction-viscous units (d).
Legend as in Table 1.

##  3 Wall-normal distributions of mean flow and Reynolds stress properties

In this section, we will first investigate the wall-normal distributions of
the mean flow, Reynolds stresses and the Reynolds stress budgets. For all the
figures presented here, profiles are plotted using a logarithmic axis for
$y^{+}$ and a linear one for $y/\delta$ to examine inner and outer layers in
more detail. Furthermore, the parameters in the inner-scaled profiles are
normalized using friction-viscous scales. Although friction-viscous scales are
not appropriate scales for the large defect case, they are still employed
because there are no suitable inner scales for comparing the small and large
defect cases (Maciel et al., 2018). The edge velocity ($U_{e}$) and $\delta$
are employed to scale the parameters in the outer region.

###  3.1 Mean flow

Figures 2a and 2b present the outer- and inner-scaled mean velocities as a
function of $y$ for all the databases. The outer-scaled mean velocity profiles
demonstrate the momentum deficit in the APG TBL cases. As the flow develops
under the effect of the APG, the defect in the mean velocity profile
increases. The profile starts to resemble velocity profiles of mixing layers
(Gungor et al., 2016) in the large defect case with an inflection point in the
middle of the boundary layer. The inner-scaled mean velocity profiles show
that the mean velocity deviates from the log law in the APG TBL cases.
Moreover, this deviation increases with increasing velocity defect.
Furthermore, friction-viscous scales progressively fail to scale the mean
velocity in the inner region as the defect increases (Gungor et al., 2016;
Maciel et al., 2018).

Figures 2c and 2d show the mean shear profiles for the DNS databases. The mean
shear profile for the experimental ZPG TBL case is not given here due to lack
of points near the wall. The mean shear distribution is important because it
directly plays a role in turbulence production and hence turbulence in the
flow. The change in the mean velocity profiles significantly affects the mean
shear distribution in the outer layer. As the defect increases, the mean shear
increases in the outer layer. More importantly, the relative magnitude of mean
shear in the outer layer with respect to the inner layer increases with
increasing velocity defect, as can be seen in figure 2d. Regarding the inner
layer, the mean shear remains fairly similar when it is normalized with
friction-viscous scales as expected, despite the direct effect of the pressure
force near the wall in the APG TBL cases. The impact of the varying mean shear
on turbulence will be further discussed in section 5.

$(a)$$(b)$ Figure 3: Reynolds stress profiles normalized with friction-viscous
units (a) and outer scales (b). Legend as in Table 1.

###  3.2 Reynolds stresses

Figure 3 presents the wall-normal distribution of the components of the
Reynolds-stress tensor for all cases. The inner scales scale the Reynolds
stresses well for the canonical flows. The $\langle u^{2}\rangle$ profiles of
the channel flow and ZPGa collapse perfectly with friction-viscous units, as
expected. The $\langle u^{2}\rangle$ levels are slightly smaller in ZPGb than
in ZPGa in the inner layer, but it is due to the lack of spatial resolution of
the probe used in the experiments (Baidya et al., 2017).

The change in the mean shear in the APG TBL (figures 2c and 2d) progressively
changes the Reynolds stress profiles. The $\langle u^{2}\rangle$ profiles are
still fairly similar for APG1 and canonical wall-bounded flows in the inner
layer, even if the scaled amplitude increases. The inner peak for $\langle
u^{2}\rangle$ still exists in APG1. However, such a similarity is not
encountered for the other components of the Reynolds stress tensor. Moreover,
figure 3b shows that the turbulent activity in the inner layer diminishes with
respect to that in the outer layer as the defect increases. The outer layer
becomes dominant as the mean shear increases in the outer layer. In the large
defect cases, APG2 and APG3, all components peak in the middle of the boundary
layer, where the mean shear has a plateau. Regarding the magnitude of the
Reynolds stresses, the levels progressively increase with increasing velocity
defect when Reynolds stresses are normalized with friction-viscous scales
because they are not appropriate scales for large velocity defect cases
(Maciel et al., 2018). It is important to state that $U_{e}$ is not
necessarily an appropriate outer scale either (Maciel et al., 2006a); however,
it conserves the order of magnitude of Reynolds stresses in the range of
velocity defects and Reynolds numbers of the cases examined here.

The Reynolds stress profiles of the current APG TBL case are consistent with
APG TBL cases in the literature. In the small defect case, there is an inner
peak for $\langle u^{2}\rangle$ and an elevated outer layer activity for all
Reynolds stress components. This has been reported for equilibrium (Skåre &
Krogstad, 1994; Lee, 2017) and non-equilibrium cases (Gungor et al., 2016) in
small defect APG TBLs. Moreover, the $y$-position and energy levels of the
inner peak matches well when the velocity defect of the cases are similar
(Kitsios et al., 2016; Tanarro et al., 2020). Regarding the large defect case,
other researchers have already reported the increasing importance of the outer
layer as the mean shear increases in the outer layer (Skåre & Krogstad, 1994;
Gungor et al., 2016; Kitsios et al., 2017).

###  3.3 Reynolds stress budgets

To understand the energy transfer mechanisms in APG TBLs, the budget of the
Reynolds stress tensor is investigated first through the transport equations
for the Reynolds stresses (equations 2).

$\begin{split}0=&-\bigg{(}\langle u_{i}u_{k}\rangle\frac{\partial
U_{j}}{\partial x_{k}}+\langle u_{j}u_{k}\rangle\frac{\partial U_{i}}{\partial
x_{k}}\bigg{)}-\frac{\partial\langle u_{i}u_{j}u_{k}\rangle}{\partial
x_{k}}\\\ &+\big{\langle}\frac{p}{\rho}(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial
x_{i}})\big{\rangle}-\frac{1}{\rho}\frac{\partial}{\partial x_{k}}(\langle
u_{i}p\rangle\delta_{jk}+\langle u_{j}p\rangle\delta_{ik})\\\
&-2\nu\langle\frac{\partial u_{i}}{\partial x_{k}}\frac{\partial
u_{j}}{\partial x_{k}}\rangle+\nu\nabla^{2}\langle u_{i}u_{j}\rangle-
U_{k}\frac{\partial\langle u_{i}u_{j}\rangle}{\partial x_{k}}\end{split}$ (2)

Here $\nu$ is viscosity and $\delta_{ik}$ is the Kronecker delta function. The
terms are in order, production, turbulent transport, pressure-strain, pressure
transport, viscous dissipation, viscous transport and mean convection.

CHZPGaAPG1APG3$(a)$

$(b)$

Figure 4: The Reynolds stress budgets. The levels and axes are normalized with
the friction-viscous scales ($a$) and outer scales ($b$). The rows are for
$\langle u^{2}\rangle$, $\langle v^{2}\rangle$, $\langle w^{2}\rangle$, and
$\langle uv\rangle$ for each figure. Production:$\bm{-}$,
Dissipation:$\bm{-}$, Pressure-strain:$\bm{-}$, Viscous diffusion:$\bm{-}$,
Turbulent transport:$\bm{-}$, Mean convection:$\bm{-}$, Pressure
transport:$\bm{-}$.

Figure 4 presents Reynolds stress budgets for canonical flows, and small- and
large-defect APG TBLs. The profiles are plotted with logarithmic and linear
axes to emphasize inner and outer layer, as mentioned before. The Reynolds
stress budget distributions show that energy follows the same main path in
both inner and outer layers, regardless of the flow as expected. Energy is
extracted from the mean flow through $\langle u^{2}\rangle$ production. Some
of this energy is transferred to $\langle v^{2}\rangle$ and $\langle
w^{2}\rangle$ through pressure strain, which is a sink term for $\langle
u^{2}\rangle$ and a source term for the other normal components except in the
very near-wall region. Energy is also transported or dissipated in all
components. Furthermore, $\langle v^{2}\rangle$ production, which is zero for
channel flows and negligible for ZPG TBLs, is a sink term that transfers
energy back to the mean flow in the outer region for APG TBLs. Regarding the
Reynolds shear stress, the production and pressure strain are almost in
balance in all cases except in the near-wall region.

The magnitude of the budget terms are very similar for CH and ZPGa for the
inner-scaled profiles. However, the levels drastically increase as the defect
increases when normalized with $u_{\tau}$, consistently with the trend we
observe in the Reynolds stress profiles, which again confirms that $u_{\tau}$
is not a proper inner scale for APG TBLs with large velocity defect.

In the inner region, as figure 4a shows, the behaviour of the Reynolds stress
budgets of the canonical flows and APG1 are almost the same for all normal
components regardless of the velocity defect. The shape of the budget profiles
is almost identical, with a minor shift in $y^{+}$. For instance, the $\langle
u^{2}\rangle$ production peak is at $y^{+}\approx 11$ and $9$ in canonical
flows and APG1, respectively. A clear difference between canonical flows and
APG1 is that the turbulent and pressure transport above $y^{+}\approx 10$ for
$\langle v^{2}\rangle$ is negligible for APG1.

As the mean shear distribution changes from APG1 to APG3, the overall
behaviour of the source and sink terms (production, pressure strain and
dissipation) in the inner layer does not considerably change except for the
aforementioned difference in magnitude. Besides the magnitude change, the main
difference is that they start to increase with $y$ above $y^{+}=30$ because
they all peak in the outer layer, as is discussed later. The change of
relative importance between the inner and outer layer affects turbulent
transport, as well. It becomes a gain term for $\langle u^{2}\rangle$ and
$\langle v^{2}\rangle$ in most of the inner layer due to the elevated
turbulent activity in the outer layer of the large defect case. Energy is
transported from the outer layer to the inner layer in APG3, unlike what
happens in the other cases.

Regarding the outer layer, as shown in the outer-scaled profiles in figure 4b,
production, pressure strain and dissipation are at high levels between
$y/\delta=0.1$ and $0.3$ in APG1, much like the Reynolds stresses themselves,
but no peak is present in the outer layer. Despite the lack of an outer peak
in APG1, a plateau between approximately $y/\delta=0.1$ and $0.3$ that is not
observed in the channel flow and the ZPG TBL exists in production and
pressure-strain terms. In the large defect case, an outer peak emerges for the
source and sink terms at around $y/\delta=0.5$. Such a behaviour has been
reported before (Skåre & Krogstad, 1994; Gungor et al., 2016; Kitsios et al.,
2017).

##  4 Spectral Analysis

The Reynolds stress and Reynolds stress budget profiles provide information
about the wall-normal distributions of energy and energy transfer, however not
about the coherent structures that carry energy or play a role in these energy
transfer mechanisms. To investigate those coherent structures, the spectral
distributions of production and pressure-strain of the Reynolds stress tensor
are analyzed using the transport equation for the two-point velocity
correlation tensor, along with the spectral distribution of energy. The reason
for using the two-point correlation equation is that the spectral information
is directly linked to the two-point correlations.

###  4.1 Methodology

We will first present the transport equation for the two-point correlation
tensor. Let $x_{i}$ and $\tilde{x_{i}}$ be the components of the coordinates
of the two points used to compute the correlations. They are defined as
follows,

$\tilde{x}_{i}=x_{i}+r_{i}$ (3)

where $r_{i}$ is the separation length between these points in direction $i$.
Further, let the velocity components at ${x_{i}}$ and $\tilde{{x_{j}}}$ be
$u_{i}$ and $\tilde{u}_{j}$. Starting from the Navier-Stokes equations, the
transport equations of the two-point correlation tensor, $R_{ij}=\langle
u_{i}\tilde{u}_{j}\rangle$, can be obtained in the following form.

$\begin{split}\bigg{\langle}\tilde{u}_{j}\frac{\partial u_{i}}{\partial
t}+u_{i}\frac{\partial\tilde{u}_{j}}{\partial
t}\bigg{\rangle}=&-\underbrace{\bigg{[}U_{k}\big{\langle}\tilde{u}_{j}\frac{\partial
u_{i}}{\partial
x_{k}}\big{\rangle}+\widetilde{U}_{k}\big{\langle}u_{i}\frac{\partial\tilde{u}_{j}}{\partial\tilde{x}_{k}}\big{\rangle}\bigg{]}}_{\text{$R_{ij}^{A}$}}-\underbrace{\bigg{[}\big{\langle}\tilde{u}_{j}u_{k}\big{\rangle}\frac{\partial
U_{i}}{\partial
x_{k}}+\big{\langle}u_{i}\tilde{u}_{k}\big{\rangle}\frac{\partial\widetilde{U}_{j}}{\partial\tilde{x}_{k}}\bigg{]}}_{\text{$R_{ij}^{P}$}}\\\
-&\underbrace{\bigg{[}\big{\langle}\tilde{u}_{j}\frac{\partial
u_{k}u_{i}}{\partial
x_{k}}\big{\rangle}+\big{\langle}u_{i}\frac{\partial\tilde{u}_{k}\tilde{u}_{j}}{\partial\tilde{x}_{k}}\big{\rangle}\bigg{]}}_{\text{$R_{ij}^{T}$}}-\underbrace{\frac{1}{\rho}\bigg{[}\big{\langle}\tilde{u}_{j}\frac{\partial
p}{\partial
x_{i}}\big{\rangle}+\big{\langle}u_{i}\frac{\partial\tilde{p}}{\partial\tilde{x}_{j}}\big{\rangle}\bigg{]}}_{\text{$R_{ij}^{\Pi}$}}\\\
+&\underbrace{\bigg{[}\nu\big{\langle}\tilde{u}_{j}\frac{\partial^{2}u_{i}}{\partial
x_{k}\partial
x_{k}}\big{\rangle}+\nu\big{\langle}u_{i}\frac{\partial^{2}\tilde{u}_{j}}{\partial\tilde{x}_{k}\partial\tilde{x}_{k}}\big{\rangle}\bigg{]}}_{\text{$R_{ij}^{\nu}$}}\end{split}$
(4)

The terms on the RHS of the equation which are labelled as $R_{ij}^{A}$,
$R_{ij}^{P}$, $R_{ij}^{T}$, $R^{\Pi}_{ij}$, $R^{\nu}_{ij}$ are advection,
production, turbulent transport, pressure and viscous terms, respectively.

The transport equations need to be simplified to perform the spectral
decompositions of the various terms. In this work, the correlations are
computed only in the streamwise and spanwise directions. Therefore there is no
separation in the wall-normal direction (eq. 5a). Since all the flows
considered here are homogeneous in the spanwise direction, the derivative of
the mean velocity with respect to $x_{3}$ and the mean velocity in $x_{3}$ is
zero (eq. 5b). Finally $u_{i}$ is independent of $\tilde{x}_{\alpha}$ since
$u_{i}$ is the velocity component at another location and this is valid for
$\tilde{u}_{i}$ and $x_{\alpha}$, too. Thus, their corresponding derivatives
are zero (eq. 5c).

$\displaystyle\tilde{x}_{2}$ $\displaystyle=x_{2}$ (5a)
$\displaystyle\frac{\partial U_{i}}{\partial x_{3}}$ $\displaystyle=U_{3}=0$
(5b) $\displaystyle\text{ }\frac{\partial u_{i}}{\partial\tilde{x}_{\alpha}}$
$\displaystyle=\frac{\partial\tilde{u}_{j}}{\partial x_{\alpha}}=0;\text{
}\alpha=1,2,3$ (5c)

We need to write equation 4 as a function of separation lengths, $r_{1}$ and
$r_{3}$, to obtain the spectral distributions. Therefore, we need to subsitute
the independent variables $x_{i}$ and $\tilde{x}_{i}$. As it is usually done,
the second substitution variable (associated with $r_{i}$) is chosen as the
average position of the two points

$\Gamma_{\alpha}=(\tilde{x}_{\alpha}+x_{\alpha})/2;\text{ }\alpha=1,3$ (6)

Let $\Phi$ be any type of two-point correlation in equation 4.2. We have the
following relationships

$\displaystyle\frac{\partial\Phi}{\partial\Gamma_{3}}=0$ (7a)
$\displaystyle\frac{\partial\Phi}{\partial\Gamma_{1}}\approx 0$ (7b)

because the term $\partial\Phi/\partial\Gamma_{3}$ is zero due to spanwise
homogeneity (eq. 7a) and the two-point correlations are assumed to vary slowly
in the streamwise direction in the case of the TBLs (eq. 7b). The latter
assumption is strong in the case of the non-equilibrium APG TBL, but it is
necessary to perform the spectral decompositions. Then we perform the
transformation for two-point correlations as shown in equation 8. Note that
the transformation does not apply to the derivatives of mean velocity in
equation 4.2 since these derivatives are assumed constant in $x_{1}$ and
$x_{3}$.

$\displaystyle\frac{\partial\Phi}{\partial x_{\alpha}}$
$\displaystyle=\frac{\partial\Phi}{\partial\Gamma_{\alpha}}\frac{\partial\Gamma_{\alpha}}{\partial
x_{\alpha}}+\frac{\partial\Phi}{\partial r_{\alpha}}\frac{\partial
r_{\alpha}}{\partial x_{\alpha}}=-\frac{\partial\Phi}{\partial r_{\alpha}}$
(8a) $\displaystyle\frac{\partial\Phi}{\partial\tilde{x}_{\alpha}}$
$\displaystyle=\frac{\partial\Phi}{\partial\Gamma_{\alpha}}\frac{\partial\Gamma_{\alpha}}{\partial\tilde{x}_{\alpha}}+\frac{\partial\Phi}{\partial
r_{\alpha}}\frac{\partial
r_{\alpha}}{\partial\tilde{x}_{\alpha}}=\frac{\partial\Phi}{\partial
r_{\alpha}}$ (8b)

Since we are interested in production and pressure-strain, we will discuss
only these two ters hereafter. The production term in equation 4 can be re-
written by taking advantage of equations 5a and 5b.

$R_{ij}^{P}=-\big{\langle}\tilde{u}_{j}u_{1}\big{\rangle}\frac{\partial
U_{i}}{\partial
x_{1}}-\big{\langle}u_{i}\tilde{u}_{1}\big{\rangle}\frac{\partial\widetilde{U}_{j}}{\partial\tilde{x}_{1}}-\big{\langle}\tilde{u}_{j}u_{2}\big{\rangle}\frac{\partial
U_{i}}{\partial
x_{2}}-\big{\langle}u_{i}\tilde{u}_{2}\big{\rangle}\frac{\partial\widetilde{U}_{j}}{\partial\tilde{x}_{2}}$
(9)

Furthermore, by assuming the derivatives of the mean velocity with respect to
$x_{1}$ and $x_{2}$ do not change in $x_{1}$ as given in equation 10,

$\begin{split}\frac{\partial\widetilde{U}_{k}}{\partial\tilde{x}_{1}}=\frac{\partial{U}_{k}}{\partial
x_{1}},\\\
\frac{\partial\widetilde{U}_{k}}{\partial\tilde{x}_{2}}=\frac{\partial{U}_{k}}{\partial
x_{2}},\end{split}$ (10)

the production term can be rewritten as follows

$R_{ij}^{P}=-\big{\langle}\tilde{u}_{j}u_{1}\big{\rangle}\frac{\partial
U_{i}}{\partial
x_{1}}-\big{\langle}u_{i}\tilde{u}_{1}\big{\rangle}\frac{\partial{U}_{j}}{\partial{x}_{1}}-\big{\langle}\tilde{u}_{j}u_{2}\big{\rangle}\frac{\partial
U_{i}}{\partial
x_{2}}-\big{\langle}u_{i}\tilde{u}_{2}\big{\rangle}\frac{\partial{U}_{j}}{\partial{x}_{2}}.$
(11)

In addition, the derivative of the mean velocity in the streamwise direction
is zero for channel flows due to the streamwise homogeneity and is negligible
in the ZPG TBLs. Therefore, the production term is simplified for the
canonical flow cases as follows,

$R_{ij}^{P}=-\big{\langle}\tilde{u}_{j}u_{2}\big{\rangle}\frac{\partial
U_{i}}{\partial
x_{2}}-\big{\langle}u_{i}\tilde{u}_{2}\big{\rangle}\frac{\partial{U}_{j}}{\partial{x}_{2}}.$
(12)

The pressure term in equation 4 is divided into two terms,

$R^{\Pi}_{ij}=R_{ij}^{d}+R_{ij}^{s},$ (13)

where $R_{ij}^{d}$ and $R_{ij}^{s}$ are pressure transport and pressure-
strain, respectively. By taking advantage of the previously introduced
relationships and some mathematical manipulations, the pressure-strain term is
written for the wall-parallel, $R_{\alpha\alpha}^{s}$, and wall-normal,
$R_{22}^{s}$, directions as follows

$\hskip 36.98866pt\text{
}R^{s}_{\alpha\alpha}=\frac{1}{\rho}\big{\langle}-{p}\frac{\partial{u}_{i}}{\partial
r_{\alpha}}+\tilde{p}\frac{\partial\tilde{u}_{j}}{\partial
r_{\alpha}}\big{\rangle};\hskip 14.22636pt\alpha=1,3$ (14)

$R^{s}_{22}=\frac{1}{\rho}\bigg{(}\big{\langle}\tilde{p}\frac{\partial{u}_{i}}{\partial
x_{2}}\big{\rangle}+\big{\langle}{p}\frac{\partial\tilde{u}_{j}}{\partial\tilde{x}_{2}}\big{\rangle}\bigg{)}.$
(15)

There are several ways to perform the decomposition of the pressure term, but
we chose this one for two reasons. Firstly we want to be consistent within the
paper, because the spectral distributions of the channel flow that we use in
the paper are obtained using the same decomposition as in Lee & Moser (2019).
The other reason is that this decomposition is consistent with many studies in
literature (Mansour et al., 1988; Mizuno, 2016) and, more importantly, with
the Reynolds stress transport equation as written in equation 3.1 when $r_{x}$
and $r_{z}$ tend to zero. In other words, by integrating the spectral
distributions of these terms over the wavenumbers, the budget of the Reynolds
stresses, which is discussed in the previous section, is obtained.

The spectral distributions of production and pressure-strain are obtained by
performing the Fourier transform of each term with respect to $r_{x}$ and
$r_{z}$. In the paper, 1D spectra are computed as a function of wavenumber
component $k_{x}$ or $k_{z}$ and 2D spectra as a function of $k_{x}$ and
$k_{z}$. The derivation above is for the 2D spectra, but a similar derivation
is performed, albeit not given here, for the 1D spectra.

We utilize temporal data to obtain the spectra in $k_{x}$. We invoke Taylor’s
frozen turbulence hypothesis to convert frequency into streamwise wavenumber

$k_{x}=\frac{2\pi f}{U_{c}}$ (16)

where $f$ is the sampling frequency and $U_{c}$ is the convection velocity
which is taken to be the local mean velocity.

In the paper, the spectra are always plotted as pre-multiplied by the
wavenumbers, and the wavenumber axes are always in logarithmic scale. For 1D
spectra, the wall-normal axis is plotted in linear scale for the outer layer
and logarithmic scale for the inner layer so that both layers are examined in
a more detailed, clear way.

CHZPGAPG1APG3$(a)$$k_{z}\phi_{uu}(k_{z},y)$$k_{x}\phi_{uu}(k_{x},y)$

$(b)$$k_{z}\phi_{vv}(k_{z},y)$$k_{x}\phi_{vv}(k_{x},y)$

$(c)$$k_{z}\phi_{ww}(k_{z},y)$$k_{x}\phi_{ww}(k_{x},y)$

$(d)$$\lambda^{+}$$\lambda^{+}$$\lambda^{+}$$\lambda^{+}$$-k_{z}\phi_{uv}(k_{z},y)$$-k_{x}\phi_{uv}(k_{x},y)$

Figure 5:  The premultiplied 1D energy (shaded), production (red) and
pressure-strain (purple) spectra of the channel flow (first column), the ZPG
TBLs (second column), APG1 (third column), and APG3 (fourth column) as a
function of $\lambda^{+}$ and $y^{+}$ for $\langle u^{2}\rangle$ (a), $\langle
v^{2}\rangle$ (b), $\langle w^{2}\rangle$ (c) and $\langle uv\rangle$ (d). For
each row the top and bottom subrows are the spanwise spectra
($k_{z}\phi(\lambda_{z},y)$) and streamwise spectra
($k_{x}\phi(\lambda_{x},y)$), respectively. The contour levels are $[0.1$
$0.3$ $0.5$ $0.7$ $0.9]$ of maxima of spectra for energy and $[0.1$ $0.3$
$0.7]$ of the maxima of spectra for production and pressure-strain. The dashed
contours indicate negative values. The horizontal and vertical dashed lines
indicate $y^{+}=10$, $100$ and $\lambda^{+}=100$ and $1000$.

CHZPGAPG1APG3$(a)$$k_{z}\phi_{uu}(k_{z},y)$$k_{x}\phi_{uu}(k_{x},y)$

$(b)$$k_{z}\phi_{vv}(k_{z},y)$$k_{x}\phi_{vv}(k_{x},y)$

$(c)$$k_{z}\phi_{ww}(k_{z},y)$$k_{x}\phi_{ww}(k_{x},y)$

$(d)$$\lambda/\delta$$\lambda/\delta$$\lambda/\delta$$\lambda/\delta$$-k_{z}\phi_{uv}(k_{z},y)$$-k_{x}\phi_{uv}(k_{x},y)$

Figure 6:  The premultiplied 1D energy (shaded contours), production (red
contours) and pressure-strain (purple contours) spectra of the channel flow
(first column), the ZPG TBLs (second column), APG1 (third column), and APG3
(fourth column) as a function of $\lambda/\delta$ and $y/\delta$ for $\langle
u^{2}\rangle$ (a), $\langle v^{2}\rangle$ (b), $\langle w^{2}\rangle$ (c) and
$\langle uv\rangle$ (d). For each row the top and bottom subrows are the
spanwise spectra ($k_{z}\phi(\lambda_{z},y)$) and streamwise spectra
($k_{x}\phi(\lambda_{x},y)$), respectively. The contour levels are $[0.1$
$0.3$ $0.5$ $0.7$ $0.9]$ of maxima of spectra for energy and $[0.3$ $0.7]$ of
the maxima of spectra for production and pressure-strain. The dashed contours
indicate negative values. The horizontal and vertical dashed lines indicate
$y/\delta=0.1$, $0.5$ and $\lambda/\delta=0.1$ and $0.75$.

The premultiplied 1D spectral distributions of energy (shaded), production
(red) and pressure-strain (purple) as a function of $y$ and wavelength
components $\lambda_{x}$ and $\lambda_{z}$ for all components of the Reynolds
stress tensor are presented using inner and outer scales in figures 5 and 6,
respectively. The 1D pressure-strain spectra are available only for the APG
TBL cases. Furthermore, $\langle v^{2}\rangle$ production is plotted only for
the APG TBL cases due to the fact it is zero for channel flows and negligible
for ZPG TBLs.

###  4.2 Energy

We start by discussing the energy spectra (shaded contours) of the channel
flow and ZPG TBLs. The spectral distribution of energy shows that energy-
carrying structures are mainly found in the inner layer in the channel flow
and ZPG TBLs, with some activity in the outer layer. A strong peak is located
at $\lambda_{z}^{+}\approx 120$ and $\lambda_{x}^{+}\approx 1000$ in the inner
layer for the $\langle u^{2}\rangle$ spectra. It is associated with the well-
known streamwise streaks in the near-wall region of canonical wall-bounded
flows. As for the outer layer, the $\langle u^{2}\rangle$ and $\langle
uv\rangle$ distributions show an outer peak in the spanwise spectra. The
streamwise spectra of CH and ZPGb do not exhibit a clear outer peak because
$\Rey_{\tau}$ is not high enough. Channel flows and ZPG TBLs with higher
$\Rey_{\tau}$ have a distinct outer peak in their streamwise spectra (Hutchins
& Marusic, 2007b; Mathis et al., 2009). This increased outer layer activity in
canonical flows, as mentioned before, is attributed to the elongated
meandering motions in the outer layer (Marusic et al., 2010a).

The energy spectra in the small defect case of the APG TBL, APG1, demonstrate
an intense turbulent activity in both inner and outer layers. The relative
importance of the turbulent activity in the outer layer with respect to the
inner layer is higher in APG1 than in the canonical flows. Energetic $\langle
u^{2}\rangle$-carrying structures are more streamwise elongated and streaky in
the inner layer than in the outer layer, and the most energetic outer layer
structures are slightly longer in the streamwise direction. The situation is
different for the other Reynolds stress components. Even though there is
energy in the inner layer, the inner peak vanishes in the streamwise and
spanwise spectra. In the outer layer (figure 6), whereas $\langle uv\rangle$
carrying structures have similar streamwise lengths as $\langle
u^{2}\rangle$-carrying ones, the $\langle v^{2}\rangle$\- and $\langle
w^{2}\rangle$-carrying structures are shorter.

The most energetic $\langle u^{2}\rangle$ structures in the inner layer of
APG1 are at $y^{+}=15$ with $\lambda_{z}^{+}$ of $120$ and $\lambda_{x}^{+}$
of approximately 650, as seen from figure 5. The spanwise wavelength and the
wall-normal location are the same as in the channel flow and the ZPG TBL. This
indicates that the wall-normal position and width of the most energetic
structures are not affected by relatively small velocity defects. Similar
values for $\lambda_{z}^{+}$ and $y^{+}$ of the inner peak have been reported
in the literature for non-equilibrium (Tanarro et al., 2020) and near-
equilibrium (Lee, 2017; Bobke et al., 2017) APG TBL cases with small velocity
defect, with shape factors varying from 1.42 to 1.74. Despite the similarity
of the wall-normal position and $\lambda_{z}^{+}$ of the inner peak, its
streamwise wavelength is smaller in APG1 than in the canonical cases where the
inner peak is at $\lambda_{x}^{+}=800$ in CH and approximately at $1000$ in
ZPGb. There are conflicting findings about $\lambda_{x}^{+}$ of the inner peak
in APG TBLs in the literature. A reduction of $\lambda_{x}^{+}$ like in the
present APG TBL has been reported by Sanmiguel Vila et al. (2020) in their TBL
cases with shape factors varying from $1.31$ to $1.51$, as the defect
increases. However, no decrease was observed by Harun et al. (2013) in their
APG TBL case with a shape factor of $1.41$ with respect to their ZPG and FPG
TBLs.

Figure 7:  The premultiplied 2D energy spectra of CH (first column), APG1
(second column), and APG3 (third column) as a function of $\lambda_{z}/\delta$
and $\lambda_{x}/\delta$ for $y/\delta=[0.1$ $0.15$ $0.3$ $0.4$ $0.5]$. The
rows are for $\langle u^{2}\rangle$, $\langle v^{2}\rangle$, $\langle
w^{2}\rangle$, and $\langle uv\rangle$. The contour levels are $[0.1$ $0.5$
$0.9]$ of the maximum of each spectra. The color is darker as $y$ increases.
The straight and dashed lines indicate $\lambda_{x}=\lambda_{z}$ and
$\lambda_{x,z}=\delta$, respectively.

In the large defect case, APG3, figure 5 shows that turbulent activity becomes
much weaker in the inner layer than in the outer layer. The inner peak in the
energy spectra of $\langle u^{2}\rangle$ vanishes completely. Furthermore, the
shape of the spectra of $\langle u^{2}\rangle$ is different from the other
flow cases. For instance, the energy found in the inner layer is at large
wavelengths. This suggests that the large scale outer layer structures might
strongly influence the inner layer in the large defect case. For the other
Reynolds stress components, the trend of the inner layer losing its importance
with respect to the outer layer continues as the defect increases from APG1 to
APG3.

The structures carrying most of the Reynolds stresses are predominantly found
in the outer layer in APG3, in the middle of the boundary layer as shown in
figure 6. The situation is the same for all components. Regarding their shape,
$\langle u^{2}\rangle$-carrying structures are more streamwise elongated than
those carrying the other components. In addition, $\langle
u^{2}\rangle$-carrying ones are the longest structures with a $\lambda_{x}$ of
$2\delta$. The most energetic structures have approximately the same
$\lambda_{z}$, $0.75\delta$, except $\langle v^{2}\rangle$-carrying ones with
$\lambda_{z}=0.25\delta$. The prominent distinction between APG3 and APG1 is
that energetic structures are located between $y/\delta=0.4-0.6$ in APG3 and
$y/\delta=0.15-0.3$ in APG1. Moreover, the shape of the 1D energy spectra of
$\langle u^{2}\rangle$ is different between the two defect cases, which
happens due to the intense inner layer activity in APG1.

To give a more complete picture of spectral properties of the energetic
structures, figure 7 presents the outer 2D spectral distributions of energy
for CH, APG1, and APG3 as a function of $\lambda_{x}/\delta$ and
$\lambda_{z}/\delta$ at several wall-normal locations (2D spectra are not
available for the ZPG TBL). The $\langle u^{2}\rangle$ and $\langle uv\rangle$
carrying structures, especially the most energetic ones, are streaky and
streamwise elongated in all flows, but they are more streamwise elongated in
the channel flow than in the APG TBL cases. For instance, the peak for
$\langle u^{2}\rangle$ spectra is at $\lambda_{x}\approx 7$-$9\delta$ in
channel flows, but it is at $\lambda_{x}\approx 2\delta$ in the APG TBL cases.
This difference can be attributed to the presence of very-large scale
structures in channel flows, which become important in high Reynolds number
flows (Marusic et al., 2010b). In contrast, $\langle v^{2}\rangle$ and
$\langle w^{2}\rangle$ carrying structure are not streamwise elongated.
Furthermore, the shape of the most energetic $\langle v^{2}\rangle$ and
$\langle w^{2}\rangle$ structures tends to follow the
$\lambda_{z}\sim\lambda_{x}$ relationship in all cases even though they are
smaller in the APG TBL cases.

The energetic structures become more self-similar in the outer layer as the
defect increases. In the channel flow, the smallest defect case, the shape of
the 2D spectra remains similar as $y$ increases, but the size of the
structures increases with respect to $\delta$ for all components. The
situation is similar in APG1, but with smaller size variations. In the large
defect case, the 2D spectra become self-similar starting from $y/\delta=0.1$.
This trend happens probably because of increasing outer layer activity as the
mean shear increases in the outer layer. As the outer layer large-scale
structures start dominating the flow and the effect of the inner layer
decreases, the structures that carry most of the energy become self-similar in
the outer layer.

Regarding the absolute levels of the spectral distributions (not shown in
figures 5 to 7 since relative levels are used), they increase when they are
normalized with $u_{\tau}$ and $U_{e}$. This is consistent with the change of
Reynolds stress profiles levels with increasing velocity defect as discussed
in section 3.2.

###  4.3 Production

After discussing the energy spectra, we continue with production spectra to
understand the behaviour of turbulence-producing structures. The $\langle
u^{2}\rangle$ and $\langle v^{2}\rangle$ production are discussed separately
since the former one is a source term and the latter one is a sink term that
is zero in channel flows and negligible in ZPG TBLs. The inner and outer
scaled $\langle u^{2}\rangle$ production spectra of figures 5a and 6a reflect
the changes that were already observed with the production profiles of figure
4. Production resides mostly in the inner layer in the case of the canonical
flows, in both layers for APG1, and mostly in the outer region for APG3. The
detailed characteristics of the $\langle u^{2}\rangle$ production spectra in
the inner and outer layers are discussed at length in Section 5.

Figure 8:  The premultiplied 2D production spectra of $\langle u^{2}\rangle$
of CH (first column), APG1 (second column), and APG3 (third column) as a
function of $\lambda_{z}/\delta$ and $\lambda_{x}/\delta$ for $y/\delta=[0.1$
$0.15$ $0.3$ $0.4$ $0.5]$. The contour levels are $[0.1$ $0.5$ $0.9]$ of the
maximum of each spectra. The color is darker as $y$ increases. The straight
and dashed lines indicate $\lambda_{x}=\lambda_{z}$ and
$\lambda_{x,z}=\delta$, respectively.

Figure 8 shows the premultiplied 2D production spectra for the $\langle
u^{2}\rangle$ component as a function of $\lambda_{x}/\delta$ and
$\lambda_{z}/\delta$ for several wall-normal locations of CH, APG1 and APG3.
Note that the $\langle uv\rangle$ spectra (figure 7, bottom row) and the
$\langle u^{2}\rangle$ production spectra have similar features because the
spectral density function of $\langle uv\rangle$ is dominant in the production
spectrum. This can be deduced from the two-point-correlation production term
of equation 4.9 with $i=j=1$. The resemblance is especially strong in the case
of the channel flow where production involving $\partial U/\partial x$ is zero
(equation 4.10). The production structures are streaky and streamwise
elongated in all cases. The shape of the spectra is similar in all cases, but
with smaller structures for the APG TBLs. Whereas the streamwise-spanwise
characteristics change with $y$ in the channel flow and APG1 to a lesser
extent, they remain the same in the large defect case as happens for the
energy spectra of figure 7.

Regarding $\langle v^{2}\rangle$ production, as it was seen from the profiles
in figure 4, it is zero in channel flow, negligible in the ZPG TBL cases and
negative in the APG TBL cases. It is found in the outer layer of both APG TBL
cases and it becomes more important with increasing velocity defect, as was
mentioned previously. The spectral distributions of figure 5b show that the
$\langle v^{2}\rangle$ production spectra are very similar to the $\langle
v^{2}\rangle$ energy spectra.

Figure 9:  The premultiplied 2D pressure-strain spectra of CH (first column),
APG1 (second column), and APG3 (third column) as a function of
$\lambda_{z}/\delta$ and $\lambda_{x}/\delta$ for $y/\delta=[0.1$ $0.15$ $0.3$
$0.4$ $0.5]$. The rows are $\langle u^{2}\rangle$, $\langle v^{2}\rangle$, and
$\langle w^{2}\rangle$. The contour levels are $[0.1$ $0.5$ $0.9]$ of the
maximum of each spectra. The color is darker as $y$ increases. The straight
and dashed lines indicate $\lambda_{x}=\lambda_{z}$ and
$\lambda_{x,z}=\delta$, respectively.

###  4.4 Inter-component energy transfer

After analyzing the turbulence-producing structures, the pressure strain
spectra, as shown in figures 5 and 6 with purple contours, are analyzed to
understand the characteristics of structures that are active in inter-
component energy transfer. Note that 1D pressure strain spectra are not
available for the canonical flows. However, 2D pressure strain spectra are
accessible for the channel flow, and are discussed further below. As
previously discussed in section 3, pressure-strain governs the inter-component
energy transfer. Since turbulence is transferred mostly from $\langle
u^{2}\rangle$ to $\langle v^{2}\rangle$ and $\langle w^{2}\rangle$, the
$\langle u^{2}\rangle$ component of the pressure strain is negative, and the
others are positive throughout the boundary layer, excluding the very near-
wall region where the $\langle v^{2}\rangle$ component is negative for both
defect cases.

In the inner layer, as figure 5 presents, most of the inter-component energy
transfer takes place with small-scale structures. In the small defect case,
$\langle u^{2}\rangle$ transfers energy from structures wider than those that
receive it for $\langle v^{2}\rangle$ and that are above those that receive it
for $\langle w^{2}\rangle$. The pressure-strain structures’ length are shorter
than most energetic structures and the structures are less streaky. The
pressure-strain spectra are qualitatively similar in the small defect case of
the APG TBL and the channel flow of Lee & Moser (2019). In the large defect
case, there is still some pressure strain in the inner layer, as happens with
production, even though most of the inter-component energy transfer takes
place in the outer layer.

Indeed, the inter-component energy transfer becomes dominant in the outer
layer and is driven by large scale structures as the mean shear increases in
the outer layer, as can be seen in figure 6. This is consistent with the
behaviour of energetic and production structures. The energy is transferred
from $\langle u^{2}\rangle$ to $\langle v^{2}\rangle$ and $\langle
w^{2}\rangle$ at similar wall-normal positions for all components, unlike what
happens in the inner layer. Whereas the spanwise wavelength of energy
transferring outer layer $\langle u^{2}\rangle$ and $\langle w^{2}\rangle$
structures are the same, $\langle v^{2}\rangle$ structures are slightly
narrower. The spanwise wavelengths and wall-normal positions where most of the
energy transfer takes place are like those of $\langle u^{2}\rangle$
production. Therefore, the outer turbulent energy is both produced and
transferred to other components by structures of similar spanwise sizes and at
similar locations. In the case of the streamwise length of the dominant
pressure-strain structures, it is shorter than that of energetic structures
for $\langle u^{2}\rangle$ and almost the same for $\langle v^{2}\rangle$ and
$\langle w^{2}\rangle$.

In the outer region, narrow structures ($\lambda_{z}<0.3\delta$ in the small
defect case and $\lambda_{z}<0.15\delta$ in the large defect case) transfer
energy predominantly from $\langle u^{2}\rangle$ to $\langle v^{2}\rangle$. In
contrast, there is no such distinction between $\langle v^{2}\rangle$ and
$\langle w^{2}\rangle$ for wide structures. Both $\langle v^{2}\rangle$ and
$\langle w^{2}\rangle$ receive a similar amount of energy, although the levels
are not given in figure 6. That energy transfer at small-scale is from
$\langle u^{2}\rangle$ to $\langle v^{2}\rangle$ is also consistent with the
fact that the energetic $\langle v^{2}\rangle$ structures have shorter
spanwise wavelengths than the energetic $\langle u^{2}\rangle$ and $\langle
w^{2}\rangle$ structures, as discussed before.

Figure 9 presents the 2D spectral distribution of pressure-strain for channel
flow, APG1 and APG3 as a function of $\lambda_{x}/\delta$ and
$\lambda_{z}/\delta$ for several wall-normal positions in the outer region.
The structures’ size notably increases in the channel flow as $y$ increases.
It increases for APG1 too, but the increase is much milder. In contrast, the
spectra are self-similar for APG3 throughout the outer layer in agreement with
the energy and production spectra. The shape of the spectra remains
qualitatively similar for each component regardless of the velocity defect.
This shows that the spectral features of the pressure-strain structures are
not significantly affected by the velocity defect, but their width and length
slightly decrease with increasing velocity defect. Pressure-strain structures
follow $\lambda_{x}\sim\lambda_{z}$.

$(a)$$(d)$$(e)$$(f)$$(b)$$(c)$ Figure 10:  The $\langle u^{2}\rangle$ ($a$)
and $\langle u^{2}\rangle$ production ($d$) profiles in the inner layer as a
function of $y$. The levels are normalized with the maximum of each profile.
The premultiplied energy ($b$,$c$) and production ($e$,$f$) spectra of
$\langle u^{2}\rangle$ as a function of $y^{+}$ and $\lambda_{z}^{+}$
($b$,$e$) and $\lambda_{x}^{+}$ ($c$,$f$) for APG1, the ZPG TBLs and the
channel flow. The contour levels are $[0.5$ $0.9]$ of the maximum of each
spectra. Colors and symbol as in Table 1.

##  5 Effects of the velocity defect on inner and outer layer turbulence

To better understand the effects of the velocity defect on energy-carrying and
energy-transferring structures, we now investigate the spectral distributions
separately in the inner and outer layers because, as mentioned above, an
intense activity in one layer hides what is happening in the other layer.
Therefore spectral distributions are now plotted separately for each layer
using their respective maxima. Furthermore, we plot spectral distributions of
energy, production and pressure-strain together because we are interested in
relative size and $y$-positions of structures with respect to each other.

Regarding the definition of the inner and outer layer, there is no commonly
accepted definition for APG TBLs (Maciel et al., 2018). For the current study,
we intend to separate the inner and outer peaks when they exist. For this
reason, we focus on the region $0<y^{+}<60$ for the inner layer and
$0.1<y/\delta<1$ for the outer region. Nevertheless, it is essential to state
that these ranges may be specific to the flows studied here.

###  5.1 Inner layer

Firstly, we investigate the inner layer of the small defect case and compare
it with those of the ZPG TBLs and channel flow. Figures 10a and 10d show the
inner-scaled $\langle u^{2}\rangle$ and $\langle u^{2}\rangle$-production
profiles in the inner layer. The parameters are normalized with their maximum
in the inner layer. The profiles of $\langle u^{2}\rangle$ collapse perfectly
for CH and ZPGa. There is a minor mismatch for ZPGb due to a decrease in
levels, as explained before. The profile of APG1 is also very similar to those
of the canonical flows. The peak is at $y^{+}=15$ in all cases. Regarding the
$\langle u^{2}\rangle$ production profiles (figure 10d), the shape is very
similar in all cases. The CH and ZPGa profiles collapse perfectly again, but
as the defect increases, the wall-normal position of the peak location
slightly decreases from $y^{+}=11$ in canonical flows to $y^{+}=9$ in APG1.

Figure 10b, 10c, 10e and 10f present the inner-scaled energy and production
spectra of $\langle u^{2}\rangle$ as a function of $\lambda^{+}$ and $y^{+}$
for the same flows. Like the profiles of $\langle u^{2}\rangle$ and $\langle
u^{2}\rangle$ production, the spectra of the three flows are similar in
friction-viscous units, the most noticeable difference being shorter
streamwise wavelengths for APG1. The difference in streamwise wavelength is
more pronounced for the production spectra (figure 10f), the production peak
is at $\lambda_{x}^{+}\approx 550$ for the channel flow and
$\lambda_{x}^{+}\approx 300$ for APG1. For the spanwise spectra (figures 10b
and 10e), the inner peak and contours scale well for the energy and production
spectra, although the production peak is located slightly below for APG1. The
aforementioned similarities in profiles and spectra suggest that the $\langle
u^{2}\rangle$ carrying and production structures are not significantly
affected by a small increase in the velocity defect, but they are definitely
shortened.

$(a)$$(b)$ Figure 11: The premultiplied energy (shaded), production (red) and
pressure-strain spectra (purple) of $\langle u^{2}\rangle$ (first column),
$\langle v^{2}\rangle$ (second column) and $\langle w^{2}\rangle$ (third
column) for the inner layer of APG1 (a) and APG3 (b). For each row the top and
bottom subrows are the spanwise spectra ($k_{z}\phi(\lambda_{z},y)$) and
streamwise spectra ($k_{x}\phi(\lambda_{x},y)$), respectively. The contour
levels are [0.1:0.2:0.9] of the maximum of each spectra in the inner layer.
The dashed contours indicate negative values. The dashed lines denote
$y^{+}=10$, $\lambda^{+}=100$ and $1000$. Figure 12:  The premultiplied 2D
energy (shaded), production (red) and pressure-strain (purple) spectra at
$y^{+}\approx 13$. The columns are the channel flow, APG1 and APG3. The rows
are $\langle u^{2}\rangle$, $\langle v^{2}\rangle$, and $\langle
w^{2}\rangle$. The contour levels are $[0.1$ $0.5$ $0.9]$ of the maximum of
each spectra.

After discussing the similarities between the canonical flows and the small
defect case, we compare the small and large defect cases of the APG TBL in the
inner layer, with this time the possibility of also comparing the pressure-
strain spectra. Figure 11 presents the energy, production and pressure-strain
spectra for all normal components of the Reynolds stress tensor for APG1 and
APG3. The outer layer is hidden, and the spectra are plotted with their
maximum in the inner layer to examine the inner layer in more detail, as
explained before. The most prominent difference between APG3 and APG1 is that
the inner peak in the energy spectra of $\langle u^{2}\rangle$ completely
vanishes in the large defect case. This is rather important because the inner
peak of $\langle u^{2}\rangle$ is partly the signature of the streaks that are
an essential part of the near-wall cycle. In contrast, both production and
pressure-strain spectral distributions of $\langle u^{2}\rangle$ for APG3
exhibit an inner peak, although the peak is not clear because the spectra are
noisy in the large defect case. As far as the shape is concerned, the shape of
production and pressure-strain spectra of $\langle u^{2}\rangle$ are similar
in both APG TBL cases. Moreover, the relative positions and dimensions of the
production and pressure-strain structures do not significantly change with
increasing velocity defect. The main difference between both defect cases is
the $y^{+}$-position and $\lambda^{+}$ of the inner peak of production and
pressure-strain spectra of $\langle u^{2}\rangle$. In APG3, the structures are
closer to the wall, and their $\lambda_{x}^{+}$ and $\lambda_{z}^{+}$ are
smaller than in APG1. As mentioned previously, this trend is expected because
of the use of friction-viscous scales (Maciel et al., 2018). Nonetheless, the
fact that the spectra remain similar indicates that the friction-viscous
length is still relevant as a length scale for production and pressure-strain
structures, at least for APG TBLs up to $H=2.6$, even if there is no
significant energy in the inner layer, as shown before, and structures become
irregular and disorganized (Maciel et al., 2017b). However, the friction-
viscous length cannot be a legit length scale in larger defect APG TBLs
considering that the friction velocity goes to zero at separation.

To complete the picture, figure 12 presents the 2D spectra of energy,
production and pressure-strain for the channel flow, APG1, and APG3 as a
function of $\lambda_{x}^{+}$ and $\lambda_{z}^{+}$ at $y^{+}\approx 13$, a
position close to the peaks of energy and production of $\langle
u^{2}\rangle$. Both energetic and production structures of $\langle
u^{2}\rangle$ are streamwise elongated, but production structures are shorter
and narrower than most energetic ones for the channel flow and the small
defect case. In addition, pressure-strain structures of $\langle u^{2}\rangle$
are much shorter and less streamwise elongated than production and energetic
structures. In the large defect case, as discussed earlier, the spectra do not
show any signs of the existence of streaks that are characteristic of the
near-wall cycle. Whereas the peak of the energy spectrum is at
$\lambda^{+}_{x}=700$-$800$ and $\lambda_{z}^{+}=120$ for channel flow and
APG1, it is at approximately $\lambda_{x}^{+}=1000$ and $\lambda_{z}^{+}=400$
for APG3. The increase in size could be due to the fact that the outer layer
large-scale wide structures dominate the inner layer. This is also consistent
with the 2D energy spectra further away from the wall that was discussed
earlier (figure 7), the structures are much more self-similar in APG3 in the
wall-normal direction. On the other hand, the production and pressure-strain
structures of $\langle u^{2}\rangle$ are shorter in APG3, but they have
similar relative sizes as in CH and APG1. Furthermore, the shape of the
production and pressure-strain spectra in all cases are alike.

As for the $\langle v^{2}\rangle$ component, the pressure-strain spectra have
a distinctive shape in all flows. They have a second peak, weaker than the
primary peak, although the peak is not clear in AGP3 due to the fact that the
spectrum is noisy. This peak is at the same $\lambda_{x}^{+}$ as the primary
peak, but $\lambda_{z}^{+}$ is considerably higher ($\lambda_{z}^{+}\approx
300$) than in the primary peak ($\lambda_{z}^{+}\approx 50$). The secondary
peak might indicate that two types of pressure-strain structures play a role
in receiving energy for $\langle v^{2}\rangle$ in the inner layer. The primary
peak’s $\lambda_{x}^{+}$ decreases with increasing velocity defect from
$\lambda_{x}^{+}\approx 300$ to $\lambda_{x}^{+}\approx 150$, while its
$\lambda_{z}^{+}$ very mildly increases.

The spectra for $\langle w^{2}\rangle$ are alike for CH and APG1 but are
different in APG3. The shape of the spectra and the size of the structures do
not considerably change as the defect increases from channel flow to APG1. The
energetic structures are bigger than the pressure-strain structures. However,
the energy spectrum for $\langle w^{2}\rangle$ become drastically different
for APG3, as happens for the $\langle u^{2}\rangle$ spectrum. The $\langle
w^{2}\rangle$-carrying structures are much broader in APG3 than in the others,
but their length is similar. The peak of the energy spectra is at
$\lambda_{z}^{+}\approx 150$ for the channel flow and APG1 but at
$\lambda_{z}^{+}\approx 350$ in APG3. On the other hand, the pressure-strain
structures behave in a similar fashion for $\langle v^{2}\rangle$ and $\langle
w^{2}\rangle$. The peak of the pressure-strain spectrum is at
$\lambda_{z}^{+}\approx 100$ but $\lambda_{x}^{+}$ decreases from
approximately $200$ to $80$ from channel flow to APG3. This behaviour is
considerably different from the behaviour of the most energetic $\langle
w^{2}\rangle$-carrying structures.

As discussed above, the velocity defect does not significantly change the
spectral characteristics of the inner layer’s energy-transferring structures.
This suggests that the well-known near-wall cycle of the canonical flows or a
turbulent regeneration mechanism with similar features to the near-wall cycle
exists in APG TBLs with small and large velocity defect. Despite the
similarity of energy-transferring structures, the signature of inner layer
streaks, which is a crucial part of the near-wall cycle, does not exist in the
$\langle u^{2}\rangle$ spectra of the large defect case, as shown in figures
11 and 12. The lack of this signature might be explained by large-scale outer
layer structures’ dominating the inner layer, as mentioned above. It is
possible that the footprints of these outer layer structures hide the
signature of the inner layer streaks in the energy spectra. A remarkable
outcome emerging from this hypothesis is that the inner layer streaks might be
able to play their part in the near-wall cycle although the inner layer is
dominated by much bigger structures.

$(a)$$(b)$ Figure 13: The premultiplied energy (shaded), production (red) and
pressure-strain spectra (purple) of $\langle u^{2}\rangle$ (first column),
$\langle v^{2}\rangle$ (second column) and $\langle w^{2}\rangle$ (third
column) for the outer layer of APG1 (a) and APG3 (b). For each row the top and
bottom subrows are the spanwise spectra ($k_{z}\phi(\lambda_{z},y)$) and
streamwise spectra ($k_{x}\phi(\lambda_{x},y)$), respectively. The contour
levels are [0.3:0.2:0.9] of the maximum of each spectra. The dashed contours
indicate negative values. The dashed lines denote $y/\delta=0.1$,
$\lambda/\delta=0.1$ and $1$.

$a)$

$b)$

Figure 14:  The premultiplied 2D energy (shaded), production (red) and
pressure-strain (purple) spectra at $y/\delta=0.15$ ($a$) and $y/\delta=0.5$
($b$). The columns are the channel flow, APG1 and APG3. The rows are $\langle
u^{2}\rangle$, $\langle v^{2}\rangle$, and $\langle w^{2}\rangle$. The contour
levels are $[0.1$ $0.5$ $0.8]$ of the maximum of each spectra.

###  5.2 Outer layer

We present now a similar analysis for the outer layer. Figure 13 displays the
energy, production and pressure-strain spectra of all normal components of the
Reynolds stress tensor for the small and large defect cases using a linear
scale for $y$ to emphasize the outer region. Moreover, the inner layer
($y/\delta<0.1$) is masked to focus on the outer layer. Some spectral
characteristics of the outer layer energy transferring structures remain
similar with increasing velocity defect even though the shape of the spectra
is different. For the $\langle u^{2}\rangle$ component, the production and
pressure-strain structures are at similar $y$-locations as the most energetic
structures. Whereas production structures have a similar size as the most
energetic structures, pressure-strain structures are slightly narrower and
considerably shorter than them in both APG TBL cases. Regarding the other
Reynolds stress components, the pressure-strain structures have a similar size
as the most energetic structures for $\langle v^{2}\rangle$ and are slightly
shorter and narrower for $\langle w^{2}\rangle$ than the most energetic ones.
Again, these relative sizes between energetic and pressure strain structures
remain similar in both defect cases.

An intriguing finding about the 1D energy spectra is that the outer layer
energy-carrying and -transferring structures appear to be independent of the
inner layer and the wall in the large defect case. This is in contrast to the
situation for the small defect case. In the latter, figures 5, 6 and 13 show
that there is considerable energy and energy transfer throughout the lower
part of the boundary layer starting from the wall up to the position of the
outer peaks in the energy spectra ($y/\delta\approx 0.25$). Moreover, the
structures’ size increases with wall-normal distance in this region. This
suggests the presence of wall-attached structures (Townsend, 1976) in the
lower part of the outer layer in APG1 like in canonical flows. It was reported
that the wall-attached structures carry most of the Reynolds shear stress, and
hence are more dominant than wall-detached structures in canonical flows (del
Álamo et al., 2006; Lozano-Durán et al., 2012). As for the large defect case,
energy and production are weak in the inner layer, and whatever activity is
present appears disconnected from the strong outer layer activity. Moreover,
wall distance does not scale the structures found in the lower part of the
outer layer. The streamwise and spanwise wavelength of the outer layer energy-
carrying and energy-transferring structures are almost the same regardless of
the wall distance in the outer layer, as shown in figures 7 to 9. These
findings suggest that the outer layer of the large defect case is populated
with detached structures. That the wall-attached structures are weaker and
less numerous in large defect APG TBLs than in ZPG TBLs has been demonstrated
before (Maciel et al., 2017b). Also, it has been reported that the wall-
detached structures in the outer layer are much more independent of the wall
in channel flows or APG TBLs than the wall-attached ones and are similar to
structures in homogeneous shear turbulence (Dong et al., 2017; Gungor et al.,
2020). Therefore, it can be concluded that the outer layer of the large defect
case is dominated by structures that are not significantly affected by the
wall, which is not the case in the small defect case. The dominance of wall-
detached structures also suggests that the outer layer of APG TBLs could
behave like free shear flows.

Despite the differences between the outer layer of the small and large defect
cases discussed above, many spectral characteristics of energy-transferring
structures remain similar regardless of the velocity defect. Figure 14 shows
the 2D spectra of the channel flow, APG1, and APG3 at $y/\delta=0.15$ and
$0.50$, which are the positions with significant energy and energy transfer in
the case of APG1 and APG3, respectively. The various 2D spectra have
strikingly similar characteristics in all flow cases at both wall-normal
positions. Their shapes, wavelength aspect ratios, and relative sizes do not
change significantly with velocity defect. At $y/\delta=0.15$, the size of all
structures slightly increases with increasing velocity defect, with the
notable exception of $\langle u^{2}\rangle$ structures that are very long in
the channel flow, reflecting the presence of VLSMs. The trend is reversed at
$y/\delta=0.5$, all structures decrease in size with increasing velocity
defect. These opposite trends possibly confirm the fact that wall-attached
structures become less dominant than wall-detached ones as the defect
increases. In the lower part of the outer layer, turbulent activity is strong
in the small defect flows, and $y$-scaled wall-attached structures are
responsible for most of this activity in canonical flows. As the defect
increases, larger wall-detached structures may start to play a bigger dynamic
role. This could explain, at least partly, the increase in structure size seen
in the 2D spectra. Although the structure size variation goes in the other
direction in the middle of the outer layer, namely a size decrease, it could
still be due to the decreasing presence of wall-attached structures. Indeed,
at that height wall-attached structures are big since they scale with $y$.
They are bigger than wall-detached ones, contrary to the situation in the
lower part of the outer layer.

Even though the dominant dynamic role in the outer layer shifts from wall-
attached to wall-detached structures as the defect increases, the striking
spectral similarities between all flows shown in figure 14 suggest that the
production and inter-component energy transfer mechanisms might still be the
same. This is consistent with the observation by Dong et al. (2017) that the
fact that dynamically relevant structures are attached to the wall in channel
flows is not the reason for their dynamic relevance. Wall attachment could
only be a consequence of their size. Therefore, in the outer region of wall-
bounded flows, the energy transfer mechanisms might remain the same no matter
if dynamically relevant structures are attached or detached to the wall.

The reason for the increase of relative importance of the outer layer with
respect to the inner layer might be the change in the mean shear since the
spectral analysis suggests that the production mechanisms remain similar in
all the considered flows. As shown previously with figures 2c and 2d, with
increasing velocity defect, mean shear significantly increases in the outer
layer (normalized with the outer scales) while it remains fairly constant in
the inner layer (normalized with the inner scales) in comparison. Consistently
with this trend, the outer layer turbulence becomes dominant as the mean shear
in the outer layer increases. This trend becomes more apparent as the defect
becomes large. Moreover, the present analysis shows that none of the cases
examined here exhibit a peak in the $\langle v^{2}\rangle$ spectra with a
frequency corresponding to a certain Strouhal number that would signal the
presence of inflection type instabilities, although the existence of such
types of instabilities in APG TBLs was reported (Elsberry et al., 2000). This
suggests that these instability mechanisms do not importantly affect the flow
or exist at all. The magnitude of the mean shear appears to be the main reason
for the increase of relative importance of the outer layer activity in APG
TBLs.

##  6 Conclusions

In the current study, we have examined the energy transfer mechanisms and
energy-carrying and -transferring structures in APG TBLs and compared them
with those of channel flows and ZPG TBLs. For this purpose, we have utilized
the spectral distributions of energy, production, and pressure-strain of the
Reynolds-stress tensor components.

The results show that spectral features of Reynolds stresses, production and
pressure strain are similar in the inner layer of APG TBLs and canonical wall-
bounded flows. The inner-scaled wall-normal location of the well-known inner
peak of $\langle u^{2}\rangle$ is the same in the small defect APG TBL case
and canonical wall-bounded flows. Moreover, energetic and production
structures in the inner layer have comparable spanwise wavelengths, although
the streamwise length of structures is shorter in the APG TBL case than in
canonical flows. When the outer layer is masked, the shape of the production
and pressure-strain spectra in the inner layer is very similar in the small
and large defect case, although there is no inner peak for the energy spectra
of $\langle u^{2}\rangle$ in the large defect case. In addition, the 2D
spectra of production and pressure-strain in the inner layer show similar
streamwise-spanwise characteristics in the channel flow and both APG TBL
cases. Therefore, energy-transferring structures and, consequently, energy
transfer mechanisms in the inner layer might be the same or very similar.

The near-wall cycle or another turbulence regeneration mechanism that works in
a similar way to the near-wall cycle appears to exist in the large defect
case. Interestingly, it would be present while the characteristic streaks are
not the dominant structures considering their signature is absent in the
energy spectra. This can be possible in two ways. The first one is that this
regeneration mechanism does not need the streaks, which is rather unlikely
considering the overall similarities between the flow cases. The other one is
that the big outer layer structures dominate the inner layer in the large
defect case in a way that the spectra do not exhibit the streak-related inner
peak, but the streaks would nonetheless be present. Interestingly, this would
mean that the streaks continue playing their role in turbulence regeneration
even if they are amid bigger and more energetic structures.

In the outer region, the wall-normal distributions of energy, production and
pressure strain spectra are different between canonical flows and APG TBLs
with small and large velocity defects. The reason for this difference is that
the relative intensity of the inner and outer layer turbulent activity
considerably changes with increasing velocity defect, from dominant inner
turbulence to dominant outer turbulence. In addition, the outer layer is
dominated by wall-detached structures in the large defect case, unlike
canonical flows or small defect APG TBLs where wall-attached structures are
responsible for carrying most of energy and energy transfer. Due to this, the
outer layer of large defect APG TBLs is much less affected by the wall than in
the small defect flows. As a result, it appears to act more like a free shear
layer than a wall-bounded flow. Despite these differences, the energy-transfer
mechanisms might still be similar in all flow cases. The 2D spectra have
strikingly similar features in the outer region of all flows, such as shape,
and wavelength aspect ratios, as well as relative sizes between energy,
production, and pressure-strain structures. This suggests that the distinction
between wall-attached structures in small defect flows and wall-detached
structures in large defect ones is not important in a dynamical sense.

Strong mean shear in the outer layer seems to be the primary reason for the
elevated outer layer activity in the APG TBL cases. The change of the wall-
normal distribution of the Reynolds stresses, production and pressure-strain
is consistent with the change of the mean shear with increasing velocity
defect. Furthermore, that the spectral analysis does not show any significant
difference between the energetic and energy transferring structures in channel
flows and APG TBLs suggests inflection-type instabilities such as the Kelvin-
Helmholtz instability either do not exist or are weak compared to other shear-
driven mechanisms. Moreover, the $\langle v^{2}\rangle$ spectra of the large-
defect APG TBL do not reveal any sign of an inflection-type instability.
However, further work is necessary to confirm the presence or absence of
inflection-type or other types of instabilities.

The overall conclusion is that the mechanisms producing turbulence and those
responsible for inter-component energy transfer may remain the same within one
layer as the velocity defect increases. What makes the large defect boundary
layer different from the small defect one seems to be mainly that the
mechanisms in the inner layer significantly decay in importance with respect
to the outer ones with increasing velocity defect. It remains to be confirmed
if this is solely due to the changes in the mean shear in each region or if
other factors are at play.

Acknowledgements

We acknowledge PRACE for awarding us access to Marconi100 at CINECA, Italy and
Calcul Québec (www.calculquebec.ca) and Compute Canada (www.computecanada.ca)
for awarding us access to Niagara HPC server. The authors would like to thank
Myoungkyu Lee and Robert D. Moser for providing their channel flow data. TRG
and AGG were supported by the research funds of Istanbul Technical University
(project numbers: MGA-2019-42227 and MDK-2018-41689). TRG and YM acknowledge
the support of the Natural Sciences and Engineering Research Council of Canada
(NSERC), project number RGPIN-2019-04194.

## References

  * del Álamo et al. (2006) del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R.D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. Journal of Fluid Mechanics 561, 329–358.
  * Baidya et al. (2021) Baidya, R., Philip, J., Hutchins, N., Monty, J.P. & Marusic, I. 2021 Spanwise velocity statistics in high-Reynolds-number turbulent boundary layers. Journal of Fluid Mechanics 913, A35.
  * Baidya et al. (2017) Baidya, R., Philip, J., Hutchins, N., Monty, J. P. & Marusic, I. 2017 Distance-from-the-wall scaling of turbulent motions in wall-bounded flows. Physics of Fluids 29 (2), 020712\.
  * Balantrapu et al. (2021) Balantrapu, N.A., Hickling, C., Alexander, W.N. & Devenport, W. 2021 The structure of a highly decelerated axisymmetric turbulent boundary layer. Journal of Fluid Mechanics 929, A9.
  * Bobke et al. (2017) Bobke, A., Vinuesa, R., Örlü, R. & Schlatter, P. 2017 History effects and near equilibrium in adverse-pressure-gradient turbulent boundary layers. Journal of Fluid Mechanics 820, 667–692.
  * Borrell et al. (2013) Borrell, G., Sillero, J.A. & Jiménez, J. 2013 A code for direct numerical simulation of turbulent boundary layers at high Reynolds numbers in BG/P supercomputers. Computers & Fluids 80, 37–43.
  * Chan et al. (2021) Chan, C.I., Schlatter, P. & Chin, R.C. 2021 Interscale transport mechanisms in turbulent boundary layers. Journal of Fluid Mechanics 921, A13.
  * Cho et al. (2018) Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. Journal of Fluid Mechanics 854, 474–504.
  * Dong et al. (2017) Dong, S., Lozano-Durán, A., Sekimoto, A. & Jiménez, J. 2017 Coherent structures in statistically stationary homogeneous shear turbulence. Journal of Fluid Mechanics 816, 167–208.
  * Elsberry et al. (2000) Elsberry, K., Loeffler, J., Zhou, M.D. & Wygnanski, I. 2000 An experimental study of a boundary layer that is maintained on the verge of separation. Journal of Fluid Mechanics 423, 227–261.
  * Griffin et al. (2021) Griffin, K.P., Fu, L. & Moin, P. 2021 General method for determining the boundary layer thickness in nonequilibrium flows. Physical Review Fluids 6 (2), 024608.
  * Gungor et al. (2017) Gungor, A.G., Maciel, Y., Simens, M.P. & Gungor, T.R. 2017 Direct numerical simulation of a non-equilibrium adverse pressure gradient boundary layer up to ${R}e_{\theta}=8000$. In 16th European Turbulence Conference, Stockholm, Sweden. European Mechanics Society.
  * Gungor et al. (2016) Gungor, A.G., Maciel, Y., Simens, M.P. & Soria, J. 2016 Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers. International Journal of Heat and Fluid Flow 59, 109–124.
  * Gungor et al. (2020) Gungor, T.R., Maciel, Y. & Gungor, A.G. 2020 Reynolds shear-stress carrying structures in shear-dominated flows. Journal of Physics: Conference Series 1522 (1), 012009.
  * Hall & Horseman (1991) Hall, P. & Horseman, N. J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. Journal of Fluid Mechanics 232, 357–375.
  * Harun et al. (2013) Harun, Z., Monty, J.P., Mathis, R. & Marusic, I. 2013 Pressure gradient effects on the large-scale structure of turbulent boundary layers. Journal of Fluid Mechanics 715, 477–498.
  * Hutchins & Marusic (2007a) Hutchins, N. & Marusic, I. 2007a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. Journal of Fluid Mechanics 579, 1–28.
  * Hutchins & Marusic (2007b) Hutchins, N. & Marusic, I. 2007b Large-scale influences in near-wall turbulence. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365 (1852), 647–664.
  * Kawata & Alfredsson (2018) Kawata, T. & Alfredsson, P.H. 2018 Inverse interscale transport of the Reynolds shear stress in plane couette turbulence. Phys. Rev. Lett. 120, 244501.
  * Kim & Adrian (1999) Kim, K.C. & Adrian, R.J. 1999 Very large-scale motion in the outer layer. Physics of Fluids 11 (2), 417–422.
  * Kitsios et al. (2016) Kitsios, V., Atkinson, C., Sillero, J.A., Borrell, G., Gungor, A.G., Jiménez, J. & Soria, J. 2016 Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer. International Journal of Heat and Fluid Flow 61, 129–136.
  * Kitsios et al. (2017) Kitsios, V., Sekimoto, A., Atkinson, C., Sillero, J.A., Borrell, G., Gungor, A.G., Jiménez, J. & Soria, J. 2017 Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation. Journal of Fluid Mechanics 829, 392–419.
  * Lee (2017) Lee, J.H. 2017 Large-scale motions in turbulent boundary layers subjected to adverse pressure gradients. Journal of Fluid Mechanics 810, 323–361.
  * Lee & Sung (2009) Lee, J.H. & Sung, H.J. 2009 Structures in turbulent boundary layers subjected to adverse pressure gradients. Journal of Fluid Mechanics 639, 101–131.
  * Lee & Moser (2015) Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $\mathit{Re}_{{\it\tau}}\approx 5200$. Journal of Fluid Mechanics 774, 395–415.
  * Lee & Moser (2019) Lee, M. & Moser, R.D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. Journal of Fluid Mechanics 860, 886–938.
  * Lozano-Durán et al. (2012) Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. Journal of Fluid Mechanics 694, 100–130.
  * Lumley (1964) Lumley, J.L. 1964 Spectral energy budget in wall turbulence. The Physics of Fluids 7 (2), 190–196.
  * Maciel et al. (2017a) Maciel, Y., Gungor, A.G. & Simens, M. 2017a Structural differences between small and large momentum-defect turbulent boundary layers. International Journal of Heat and Fluid Flow 67, 95–110.
  * Maciel et al. (2006a) Maciel, Y., Rossignol, K.S. & Lemay, J. 2006a Self-similarity in the outer region of adverse-pressure-gradient turbulent boundary layers. AIAA Journal 44 (11), 2450–2464.
  * Maciel et al. (2006b) Maciel, Y., Rossignol, K.S. & Lemay, J. 2006b A study of a turbulent boundary layer in stalled-airfoil-type flow conditions. Experiments in fluids 41 (4), 573–590.
  * Maciel et al. (2017b) Maciel, Y., Simens, M.P. & Gungor, A.G. 2017b Coherent structures in a non-equilibrium large-velocity-defect turbulent boundary layer. Flow, Turbulence and Combustion 98 (1), 1–20.
  * Maciel et al. (2018) Maciel, Y, Wei, T., Gungor, A.G. & Simens, M.P. 2018 Outer scales and parameters of adverse-pressure-gradient turbulent boundary layers. Journal of Fluid Mechanics 844, 5–35.
  * Mansour et al. (1988) Mansour, N.N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. Journal of Fluid Mechanics 194, 15–44.
  * Marquillie et al. (2011) Marquillie, M., Ehrenstein, U. & Laval, J.P. 2011 Instability of streaks in wall turbulence with adverse pressure gradient. Journal of Fluid Mechanics 681, 205–240.
  * Marusic et al. (2010a) Marusic, I., Mathis, R. & Hutchins, N. 2010a High Reynolds number effects in wall turbulence. International Journal of Heat and Fluid Flow 31 (3), 418–428.
  * Marusic et al. (2010b) Marusic, I., McKeon, B.J., Monkewitz, P. A., Nagib, H. M., Smits, A.J. & Sreenivasan, K.R. 2010b Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues. Physics of Fluids 22 (6), 065103\.
  * Mathis et al. (2009) Mathis, R., Monty, J. P., Hutchins, N. & Marusic, I. 2009 Comparison of large-scale amplitude modulation in turbulent boundary layers, pipes, and channel flows. Physics of Fluids 21 (11), 111703.
  * Mizuno (2016) Mizuno, Y. 2016 Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. Journal of Fluid Mechanics 805, 171–187.
  * Rahgozar & Maciel (2012) Rahgozar, S. & Maciel, Y. 2012 Statistical analysis of low- and high-speed large-scale structures in the outer region of an adverse pressure gradient turbulent boundary layer. Journal of Turbulence 13, N46.
  * Sanmiguel Vila et al. (2020) Sanmiguel Vila, C., Vinuesa, R., Discetti, S., Ianiro, A., Schlatter, P. & Örlü, R. 2020 Separating adverse-pressure-gradient and Reynolds-number effects in turbulent boundary layers. Phys. Rev. Fluids 5, 064609.
  * Sanmiguel Vila et al. (2020) Sanmiguel Vila, C., Vinuesa, R., Discetti, S., Ianiro, A., Schlatter, P. & Örlü, R. 2020 Experimental realisation of near-equilibrium adverse-pressure-gradient turbulent boundary layers. Experimental Thermal and Fluid Science 112, 109975.
  * Schatzman & Thomas (2017) Schatzman, D.M. & Thomas, F.O. 2017 An experimental investigation of an unsteady adverse pressure gradient turbulent boundary layer: embedded shear layer scaling. Journal of Fluid Mechanics 815, 592–642.
  * Sillero et al. (2013) Sillero, J.A., Jiménez, J. & Moser, R.D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to $\delta^{+}\approx$ 2000\. Physics of Fluids 25 (10), 105102.
  * Simens et al. (2009) Simens, M.P., Jiménez, J., Hoyas, S. & Mizuno, Y. 2009 A high-resolution code for turbulent boundary layers. Journal of Computational Physics 228 (11), 4218–4231.
  * Skåre & Krogstad (1994) Skåre, P.E. & Krogstad, P. 1994 A turbulent equilibrium boundary layer near separation. Journal of Fluid Mechanics 272, 319–348.
  * Skote & Henningson (2002) Skote, M. & Henningson, D. S. 2002 Direct numerical simulation of a separated turbulent boundary layer. Journal of Fluid Mechanics 471, 107–136.
  * Smits et al. (2011) Smits, A.J., McKeon, B.J. & Marusic, I. 2011 High–Reynolds number wall turbulence. Annual Review of Fluid Mechanics 43, 353–375.
  * Tanarro et al. (2020) Tanarro, Á., Vinuesa, R. & Schlatter, P. 2020 Effect of adverse pressure gradients on turbulent wing boundary layers. Journal of Fluid Mechanics 883, A8.
  * Townsend (1976) Townsend, A. A. 1976 The structure of turbulent shear flow. Cambridge university press.