# Generalised quasilinear approximations of turbulent channel flow: Part 2.
Spanwise scale interactions

Carlos G. Hernández1<EMAIL_ADDRESS>Qiang Yang2  Yongyun Hwang1  1
Department of Aeronautics, Imperial College London, South Kensington, London
SW7 2AZ, UK 2 State Key Laboratory of Aerodynamics, China Aerodynamics
Research and Development Centre, Mianyang 621000, PR China

###### Abstract

Continuing from Part 1 (Hernández _et al._ , _arXiv:2108.12395_ , 2021), a
generalized quasilinear (GQL) approximation is studied in turbulent channel
flow using a flow decomposition defined with spanwise Fourier modes: the flow
is decomposed into a set of low-wavenumber spanwise Fourier modes and the rest
high-wavenumber modes. This decomposition leads to the nonlinear low-
wavenumber group that supports the self-sustaining process within the given
integral length scales, whereas the linearised high-wavenumber group is not
able to do so, unlike the GQL models in Part 1 which place a minimal
mathematical description for the self-sustaining process across all integral
scales. Despite the important physical difference, it is shown that the GQL
models in this study share some similarities with those in Part 1: i.e. the
reduced multi-scale behaviour and anisotropic turbulent fluctuations.
Furthermore, despite not being able to support the self-sustaining process in
the high-wavenumber group, the GQL models in the present study are found to
reproduce some key statistical features in the high-wavenumber group solely
through the ‘scattering’ mechanism proposed by previous studies. Finally,
using the nature of the GQL approximation, a set of numerical experiments
suppressing certain triadic nonlinear interactions are further carried out.
This unveils some key roles played by the certain types of triadic
interactions including energy cascade and inverse energy transfer in the near-
wall region. In particular, the inhibition of inverse energy transfer in the
spanwise direction leads to suppression of the near-wall positive turbulent
transport at large scales.

##  1 Introduction

The quasilinear (QL) and generalized quasilinear (GQL) approximations have
recently been proposed in various forms for several turbulent flows. Common to
all various forms, this approach introduces a decomposition of the given flow
into two groups: one in which nonlinear terms are kept, and the other in which
all self-interactions are ignored or suitably modelled. The resulting
equations for the first are unchanged from or close to the original, while
those for the second become equivalent to a linearisation around the first
group often with an additional model (e.g. stochastic forcing). The early work
may be found in Malkus (1954, 1956) and Herring (1963, 1964, 1966) where the
equations for the second group are closed using the ‘marginal stability’
ansatz. In the last two decades, several variants of the QL framework have
been proposed for many different flows with various types of models for the
self-interaction term of the second group (e.g. stochastic forcing, eddy
viscosity, etc): for example, stochastic structural stability theory (S3T)
(Farrell & Ioannou, 2003, 2007, 2012), direct statistical simulation (DSS)
(Marston et al., 2008; Tobias & Marston, 2013), self-consistent approximation
for linearly unstable flows (Mantič-Lugo et al., 2014; Mantič-Lugo & Gallaire,
2016), minimal quasilinear approximation augmented with an eddy-viscosity
model (Hwang & Eckhardt, 2020; Skouloudis & Hwang, 2021), restricted nonlinear
model (RNL) (Thomas et al., 2014, 2015; Farrell et al., 2017; Pausch et al.,
2019; Hernández & Hwang, 2020) and generalised quasilinear approximations
(GQL) (Constantinou, 2015; Marston et al., 2016; Tobias & Marston, 2017).

Continuing from Part 1 (Hernández, Yang & Hwang, 2021), the starting point of
the present study is the RNL model, which we have referred to as the QL model.
The important advantage of this model is that it can capture the so-called
‘self-sustaining process’ (Hamilton et al., 1995; Waleffe, 1997), i.e. the
two-way interaction between ‘streamwise elongated’ structure of streamwise
velocity (streaks) and its ‘streamwise wavy’ instability involving cross-
streamwise velocities (waves and rolls), in a minimal manner (Thomas et al.,
2014, 2015; Pausch et al., 2019; Hernández & Hwang, 2020) – in the QL model,
the elongated streaks are captured by the nonlinear streamwise mean equations
and the streamwise wavy instability is by the linearised equations. The recent
examination of this model in uniform shear turbulence (Hernández & Hwang,
2020) revealed that the inhibition of the streamwise energy cascade in the QL
model by construction results in highly elevated spectral energy intensity
residing only at the streamwise integral length scale, and only a small number
of streamwise Fourier modes consequently remain active Thomas et al. (2014,
2015); Farrell et al. (2016); Tobias & Marston (2017). While the QL model
captures the most fundamental physical mechanism in wall-bounded turbulence
reasonably well (especially at low Reynolds numbers), it was found that the
approximation yields some non-negligible damages to the slow pressure. This
consequently inhibits the related pressure strain which transfers the
turbulent kinetic energy produced at the streamwise component to the cross-
streamwise components, explaining the typical anisotropic structures in the QL
model throughout the entire wavenumber space.

In Part 1 (Hernández et al., 2021), the QL model was extended by applying the
GQL approximation to the streamwise direction in turbulent channel flow at
$Re_{\tau}\simeq 1700$ to explore a route to improvement of the QL model. The
GQL approximation employs the flow decomposition in the QL model in a more
flexible manner: the nonlinear equations for the streamwise mean flow is
replaced with nonlinear equations for a set of low-wavenumber streamwise
Fourier modes (i.e. the low-wavenumber group), and the rest of the flow (i.e.
the high-wavenumber group) is described by the linearised equations about the
set of low-wavenumber streamwise Fourier modes. This is a direct extension of
the previous QL model (Thomas et al., 2014, 2015; Farrell et al., 2017; Pausch
et al., 2019; Hernández & Hwang, 2020), which offers the dynamics of the
energy-containing streamwise waves in a minimal manner. In Part 1 (Hernández
et al., 2021), the GQL approximation was found to improve the QL model
effectively. By incorporation of only a few more streamwise Fourier modes into
the low-wavenumber group, the GQL model was able to recover the self-similar
scaling with the distance from the wall of the turbulence statistics and
spectra. This was associated with the mean-fluctuation interaction improved by
the enhanced energy transfer in the streamwise wavenumber space. Furthermore,
it was found that this energy transfer from the low to the high-wavenumber
group in the GQL model depends on the neutrally stable leading Lyapunov
spectrum of the linearised equations for the high-wavenumber group. In
particular, if the threshold wavenumber distinguishing the two groups is
sufficiently high, this energy transfer is inhibited due to the linear nature
of the equations for the high-wavenumber group.

The GQL model employed in Part 1 provides useful physical insights into the
modelling of wall-bounded turbulence in the context of the self-sustaining
process. However, if the Reynolds number is sufficiently high, the self-
sustaining processes emerge at all (spanwise) integral length scales that vary
continuously with the distance from the wall (Flores & Jiménez, 2010; Hwang &
Cossu, 2010a, 2011; Hwang, 2015; Hwang & Bengana, 2016). There have been a
number of previous investigations on the interactions between the related
energy-containing fluid motions in the near-wall and outer regions (Hutchins &
Marusic, 2007; Mathis et al., 2009; Duvvuri & McKeon, 2015; Agostini &
Leschziner, 2016; Zhang & Chernyshenko, 2016). In these studies, the velocity
field is often decomposed into large and small scales, and their interactions
have been modelled and explored in terms of superposition and amplitude
modulation in the near-wall region. However, in general, such interactions are
expected to exist across almost every wall-normal locations, given the
existence of the self-sustaining processes across all integral length scales
in the form of the ‘attached eddies’ of Townsend (1976). Recent investigations
on the statistical features in the spectral energy transfer revealed that the
interaction dynamics appears to be highly complex (Kawata & Alfredsson, 2018;
Cho et al., 2018; Lee & Moser, 2019), although there were some progresses made
for an idealised system in which there are only two integral length scales
(Doohan et al., 2021b, a). The newly-found scale interactions include
mediation of the energy cascade from given integral scale by larger energy-
containing motions (Cho et al., 2018), small-scale turbulence production
driven by energy cascade from larger scales (Doohan et al., 2021b), the near-
wall inverse energy transfer that forms large-scale near-wall inactive motions
of Townsend (1976) (Cho et al., 2018; Doohan et al., 2021b, a), and the mean-
fluctuation interactions across all integral scales (Doohan et al., 2021a).

Figure 1: Set of triadic interactions. In the GQL approximation, three
interactions (a,c,e) are retained out of six in total. The other three triads
are discarded (Marston et al., 2016).

As was briefly discussed in Part 1, the GQL approximation suppresses
particular triadic wave interactions between the low and high wavenumber
groups. This was sketched in figure 1, where the GQL approximations ignore
those in (b), (d) and (f). Similarly, this feature can further be exploited to
study the scale interactions in wall-bounded turbulence: for example, the
other interactions such as (c) can also be suppressed to understand the effect
of the inverse energy transfer. From this perspective, a flexible utilisation
of the GQL approximation may perhaps provide a unique interventional tool to
explore these scale interactions.

The objective of the present study is to explore the scale interactions in
wall-bounded turbulence using the GQL approximation. For this purpose, unlike
Part 1, the low wavenumber-mode group will be composed of the plane Fourier
modes for $k_{z}\leq k_{z,c}$ ($k_{z}$ is the spanwise Fourier wavenumber and
$k_{z,c}$ the corresponding threshold wavenumber for the decomposition). The
spanwise length scale has been understood to represent the size of coherent
structures underpinned by the self-sustaining process, as the spanwise Fourier
modes well represent the streaks highly elongated in the streamwise directions
and the related quasi-streamwise vortical structures (e.g. Hwang, 2015; Cho et
al., 2018). The proposed GQL approximation will then be first examined to
present its effect in the spanwise direction. Further to this, a set of
numerical experiments, which deliberately suppress the terms presented in
figure 1, will be performed to understand the scale interaction in wall-
bounded turbulence.

The paper is organized as follows. The GQL approximation together with its
spectral energy budget is briefly formulated in section 2 for completeness. In
section 3, the statistics and spectra of the GQL model with spanwise threshold
wavenumber for channel flow are compared to those of a wall-resolved LES. The
energy-budget spectra are also presented here with a detailed analysis to
explain the statistical features of the GQL model in the spanwise direction.
Comparison of the GQL in the present study is subsequently made with that in
Part 1, and various numerical experiments suppressing certain interactions are
carried out in section 4, highlighting the use of the GQL-approximation
oriented idea to analyze scale interactions. Finally, the paper concludes in
section 5 with some remarks.

##  2 Problem formulation

###  2.1 Generalised quasilinear approximation

We consider a pressure-driven plane channel flow, in which the density and
kinematic viscosity are given by $\rho$ and $\nu$, respectively. The time is
denoted by $t$, and the space is by $\mathbf{x}=(x,y,z)$ with $x$, $y$ and $z$
being the streamwise, wall-normal and spanwise directions, respectively. The
lower and upper walls of the channel are set to be located at $y=0,2h$. For
the GQL approximation, the velocity $\bf u$ is decomposed into two groups
using a discrete Fourier transform in the wall-parallel directions:

$\mathbb{u}=\mathbb{U}_{l}+\mathbb{u}_{h},$ (1a) where
$\mathbb{U}_{l}=\sum_{n=-M_{z,F}}^{M_{z,F}}\sum_{m=-M_{x,F}}^{M_{x,F}}\hat{\mathbb{u}}_{m,n}e^{i(mk_{x,0}x+nk_{z,0}z)}$
(1b)

and $\mathbf{u}_{h}$ is given from (1a). Here, $\hat{\mathbb{u}}_{m,n}$ is the
discrete Fourier mode of the velocity, $k_{x,0}$ and $k_{z,0}$ are the
fundamental streamwise and spanwise wavenumbers for the given horizontal
domain, and $M_{x,F}$ and $M_{z,F}$ define the threshold streamwise and
spanwise wavenumbers for the decomposition such that $k_{x,c}=M_{x,F}k_{x,0}$
and $k_{z,c}=M_{z,F}k_{z,0}$. Given the scope of the present study, the main
parameter of interest is $k_{z,c}$ (or $M_{z,F}$), while keeping $M_{x,F}$ to
be identical to the number of the streamwise Fourier modes used in the full
simulation (see §2.3 for further details).

With the decomposition in (1), two related projection operators are defined as

$\mathcal{P}_{l}[\mathbb{u}]\equiv\mathbb{U}_{l},~{}~{}\mathcal{P}_{h}[\mathbb{u}]\equiv\mathbb{u}-\mathbb{U}_{l}=\mathbb{u}_{h},$
(2)

with the following properties:

$\mathcal{P}_{l}[\cdot]+\mathcal{P}_{h}[\cdot]=\mathcal{I}[\cdot],$ (3a)
$\mathcal{P}_{l}[\mathcal{P}_{l}[\cdot]]=\mathcal{P}_{l}[\cdot],~{}\mathcal{P}_{h}[\mathcal{P}_{h}[\cdot]]=\mathcal{P}_{h}[\cdot],$
(3b)
$\mathcal{P}_{l}[\mathcal{P}_{h}[\cdot]]=\mathcal{P}_{h}[\mathcal{P}_{l}[\cdot]]=\mathbf{0},$
(3c)

where $\mathcal{I}[\cdot]$ is the identity operator. Using the definition and
the properties listed in (2) and (3), the Navier-Stokes equations are first
projected onto the $\mathcal{P}_{l}$ (or low wavenumber) and $\mathcal{P}_{h}$
(or high wavenumber) subspaces. The subsequent linearisation of the equations
for $\mathbf{u}_{h}$ about $\mathbf{U}_{l}$ leads to the GQL system of
interest, i.e.

$\displaystyle\partialderivative{\mathbb{U}_{l}}{t}+\mathcal{P}_{l}[(\mathbb{U}_{l}\cdot\nabla)\mathbb{U}_{l}]=-\frac{1}{\rho}\mathbb{\nabla}P_{l}+\nu\mathbb{\nabla}^{2}\mathbb{U}_{l}+\mathcal{P}_{l}[\mathbb{\nabla}\cdot\bm{\tau}_{SGS}]-\mathcal{P}_{l}\left[\left(\textbf{u}_{h}\cdot\nabla\right)\textbf{u}_{h}\right],$
(4a) and
$\partialderivative{\mathbb{u}_{h}}{t}+\mathcal{P}_{h}[(\mathbb{u}_{h}\cdot\mathbb{\nabla})\mathbb{U}_{l}]+\mathcal{P}_{h}[(\mathbb{U}_{l}\cdot\mathbb{\nabla})\mathbb{u}_{h}]=-\frac{1}{\rho}\mathbb{\nabla}p_{h}+\nu\mathbb{\nabla}^{2}\mathbb{u}_{h}+\mathcal{P}_{h}[\mathbb{\nabla}\cdot\bm{\tau}_{SGS}],$
(4b)

where $P_{l}$ and $p_{h}$ are given to enforce
$\mathbb{\nabla}\cdot\mathbb{U}_{l}=0$ and
$\mathbb{\nabla}\cdot\mathbb{u}_{h}=0$, respectively, with $p=P_{l}+p_{h}$. As
in Part 1, the terms
$\mathcal{P}_{l}\left[\left(\mathbb{U}_{l}\cdot\nabla\right)\mathbb{u}_{h}\right]$
and
$\mathcal{P}_{l}\left[\left(\mathbb{u}_{h}\cdot\nabla\right)\mathbb{U}_{l}\right]$
are neglected in (4a). The numerical simulations in this study are performed
using LES. Therefore, the subgrid-scale stress (SGS) tensor, given by
$\bm{\mathbf{\tau}}_{SGS}=-(\mathbb{u}\mathbb{u}-\overline{\mathbb{u}\mathbb{u}})$,
appears in (4), where the overbar $(\overline{\cdot})$ denotes the application
of the grid filter given by the numerical discretisation of the equations (see
§2.3). The SGS tensor for the present LES here employs a mixing-length type
model
$\bm{\mathbf{\tau}}_{SGS}=\nu_{t}(\nabla\mathbb{u}+\nabla\mathbb{u}^{T})$,
where the eddy viscosity $\nu_{t}$ is determined with the Vreman model
(Vreman, 2004). Further details are provided in Part 1 (Hernández et al.,
2021).

###  2.2 Spectral energy budget

To analyse the turbulence statistics and spectra of the given GQL and original
systems, the Reynolds decomposition of the velocity $\mathbf{u}=(u,v,w)$ is
employed: i.e.

$\mathbf{u}=\mathbf{U}+\mathbf{u}^{\prime},$ (5)

in which $\mathbf{U}(\equiv\langle\mathbf{u}\rangle_{x,z,t})=(U(y),0,0)$ is
the mean velocity with $\langle\cdot\rangle_{x,z,t}$ being an average in $t$-,
$x$\- and $z$-directions. Similarly to Part 1, this decomposition is only
introduced for the statistical analysis of LES and GQL models, and it should
not be confused with the one in (1) introduced for the GQL approximations.
Using (1) and (5), the turbulent velocity fluctuation is further decomposed
into the low and high wavenumber components as in (1a):

$\mathbb{u}^{\prime}=\mathbb{u}_{l}+\mathbb{u}_{h}.$ (6)

Case | $Re$ | $Re_{\tau}$ | $\Delta_{x}^{+}$ | $\Delta_{z}^{+}$ | $M_{x,F}$ | $M_{z,F}$ | $\lambda_{z,c}/h$ | $\lambda_{z,c}^{+}$ | $N_{x}\times N_{y}\times N_{z}$
---|---|---|---|---|---|---|---|---|---
LES | 66667 | 1974 | 64.6 | 32.3 | 384 | 96 | - | - | $1152\times 169\times 288$
GQLZ1 | 66667 | 1816 | 59.4 | 29.7 | 384 | 1 | 3.14 | 5702 | $1152\times 169\times 288$
GQLZ5 | 66667 | 1945 | 63.6 | 31.8 | 384 | 5 | 0.63 | 1221 | $1152\times 169\times 288$
GQLZ25 | 66667 | 1958 | 64.0 | 32.0 | 384 | 25 | 0.13 | 246 | $1152\times 169\times 288$
GQLZ64 | 66667 | 1978 | 64.7 | 32.3 | 384 | 64 | 0.05 | 97 | $1152\times 169\times 288$
TRIAZ25 | 66667 | 1981 | 64.8 | 32.4 | 384 | 25 | 0.13 | 249 | $1152\times 169\times 288$
TRIBZ25 | 66667 | 1860 | 60.8 | 30.4 | 384 | 25 | 0.13 | 234 | $1152\times 169\times 288$
TRICZ25 | 66667 | 1721 | 56.3 | 28.1 | 384 | 25 | 0.13 | 216 | $1152\times 169\times 288$
TRIDZ25 | 66667 | 2066 | 67.6 | 33.8 | 384 | 25 | 0.13 | 259 | $1152\times 169\times 288$
Table 1: Simulation parameters in the present study. The domain size in the
$x$-, $y$\- and $z$ directions is $L_{x}/h=8\pi$, $L_{y}/h=2$ and
$L_{z}/h=\pi$, respectively. Here, $Re=U_{0}h/\nu$ and
$Re_{\tau}=u_{\tau}h/\nu$, where $U_{0}$ and $u_{\tau}$ are the centerline and
wall shear velocities, respectively. The grid spacings in the $x$\- and
$z$-directions are $\Delta_{x}^{+}$ and $\Delta_{z}^{+}$ (after aliasing).
$\lambda_{z,c}$ is the threshold spanwise wavelength. $N_{x}$, $N_{y}$,
$N_{z}$ are the number of grid points in the $x$-, $y$\- and $z$-directions,
respectively.

As in Part 1, the energy balance in the Fourier space for the GQL
approximation is given by

$\displaystyle
0=\underbrace{\left\langle\Real\left\\{-{\widehat{u^{\prime}}}^{*}(k)\widehat{v^{\prime}}(k)\>\frac{\textup{d}U}{\textup{d}y}\right\\}\right\rangle_{r^{\perp},t}}_{\widehat{P}\left(y,k\right)}+\underbrace{\left\langle-\nu\frac{\partial{{\widehat{u_{i}^{\prime}}}^{*}(k)}}{\partial{x_{j}}}\frac{\partial{{\widehat{u_{i}^{\prime}}}(k)}}{\partial{x_{j}}}\>\right\rangle_{r^{\perp},t}}_{\widehat{\varepsilon}\left(y,k\right)}$
$\displaystyle+\underbrace{\left\langle\Real\left\\{-{\widehat{u_{i}^{\prime}}}^{*}(k)\left(\frac{\partial}{\partial{x_{j}}}\left(\widehat{\tau_{ij,SGS}^{\prime}}(k)\right)\right)\>\right\\}\right\rangle_{r^{\perp},t}}_{\widehat{\varepsilon}_{SGS}\left(y,k\right)}+\underbrace{\left\langle\Real\left\\{\frac{\textup{d}}{\textup{d}y}\left(\frac{\widehat{p^{\prime}}(k){\widehat{v^{\prime}}}^{*}(k)}{\rho}\right)\right\\}\right\rangle_{r^{\perp},t}}_{\widehat{T}_{p}\left(y,k\right)}$
$\displaystyle+\underbrace{\left\langle\Real\left\\{-{\widehat{u_{i}^{\prime}}}^{*}(k)\left(\frac{\partial}{\partial{x_{j}}}\left(\widehat{u_{i}^{\prime}u_{j}^{\prime}}(k)-\mathcal{P}_{h}\left[\widehat{u_{h,i}^{\prime}u_{h,j}^{\prime}}(k)\right]-\mathcal{P}_{h}\left[\widehat{u_{l,i}^{\prime}u_{l,j}^{\prime}}(k)\right]\right.\right.\right.\right.}_{\widehat{T}_{turb}\left(y,k\right)}$
$\displaystyle\underbrace{\left.\left.\left.\left.-\mathcal{P}_{l}\left[\widehat{u_{l,i}^{\prime}u_{h,j}^{\prime}}(k)\right]-\mathcal{P}_{l}\left[\widehat{u_{h,i}^{\prime}u_{l,j}^{\prime}}(k)\right]\right)\right)\>\right\\}\right\rangle_{r^{\perp},t}}_{\widehat{T}_{turb}\left(y,k\right)}+\underbrace{\left\langle\nu\frac{\textup{d}^{2}{\widehat{e}(k)}}{\textup{d}{y}^{2}}\right\rangle_{r^{\perp},t}}_{\widehat{T}_{\nu}\left(y,k\right)},$
(7)

for $i,j=1,2,3$. Here, $(x_{1},x_{2},x_{3})=(x,y,z)$,
$(u_{1}^{\prime},u_{2}^{\prime},u_{3}^{\prime})=(u^{\prime},v^{\prime},w^{\prime})$,
$(\widehat{\cdot})$ denotes the Fourier-transformed coefficient in the
$r$-direction, where $r(=x~{}\mathrm{or}~{}z)$ is the streamwise or spanwise
coordinate with $r^{+}(=z~{}\mathrm{or}~{}x)$ being the wall-parallel
direction orthogonal to $r$, and $k(=k_{x}~{}\mathrm{or}~{}k_{z})$ the spatial
wavenumber in the $r$-direction. Also,
$\widehat{e}(k)=(|{\widehat{u^{\prime}}(k)}|^{2}+|{\widehat{v^{\prime}}(k)}|^{2}+|{\widehat{w^{\prime}}(k)}|^{2})/2$,
the superscript $(\cdot)^{*}$ indicates the complex conjugate, and
$\Real\left\\{\hskip 2.84526pt\cdot\hskip 2.84526pt\right\\}$ the real part.
The terms on the right-hand side are the rate of turbulence production
$\widehat{P}(y,k)$, viscous dissipation $\widehat{\varepsilon}(y,k)$, SGS
dissipation $\widehat{\varepsilon}_{SGS}(y,k)$, pressure transport
$\widehat{T}_{p}(y,k)$, turbulent transport $\widehat{T}_{turb}(y,k)$ and
viscous transport $\widehat{T}_{\nu}(y,k)$, respectively.

((a)) $u_{rms}^{+}(y^{+})$

((b)) $v_{rms}^{+}(y^{+})$

((c)) $U^{+}(y^{+})$

((d)) $\langle u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}(y^{+})$

Figure 2: First- and second-order turbulence statistics for the present LES,
DNS at $Re_{\tau}=2003$ (Hoyas & Jiménez, 2008), GQLZ1, GQLZ5 and GQLZ25
cases: (a) $u_{rms}^{+}(y^{+})$; (b) $v_{rms}^{+}(y^{+})$; (c) $U^{+}(y^{+})$;
(d) $\langle u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}(y^{+})$.

Equation (2.2) can further be split into each component. In this case, the
energy transports by pressure strain subsequently appear as

$\displaystyle\widehat{\Pi}_{x}(y,k)=\left\langle\Real\left\\{\frac{\widehat{p^{\prime}}(k)}{\rho}\partialderivative{{\widehat{u^{\prime}}}^{*}(k)}{x}\right\\}\right\rangle_{r^{\perp},t},\quad\widehat{\Pi}_{y}(y,k)=\left\langle\Real\left\\{\frac{\widehat{p^{\prime}}(k)}{\rho}\partialderivative{{\widehat{v^{\prime}}}^{*}(k)}{y}\right\\}\right\rangle_{r^{\perp},t},$
$\displaystyle\widehat{\Pi}_{z}(y,k)=\left\langle\Real\left\\{\frac{\widehat{p^{\prime}}(k)}{\rho}\partialderivative{{\widehat{w^{\prime}}}^{*}(k)}{z}\right\\}\right\rangle_{r^{\perp},t},$
(8)

where $\widehat{\Pi}_{x}$, $\widehat{\Pi}_{y}$ and $\widehat{\Pi}_{z}$ are
one-dimensional spectra of the streamwise, wall-normal and spanwise components
of pressure strain, respectively. From the incompressibility of the given
fluid, the pressure strain terms satisfy the following relation

$\widehat{\Pi}_{x}(y,k)+\widehat{\Pi}_{y}(y,k)+\widehat{\Pi}_{z}(y,k)=0,$ (9)

and a further discussion on this can be found in previous studies (Mizuno,
2016; Cho et al., 2018; Lee & Moser, 2019; Hernández & Hwang, 2020; Hernández
et al., 2021).

###  2.3 Numerical simulations

A large eddy simulation (LES) is first carried out with the same numerical
solver used in Part 1 (Hernández & Hwang, 2020). The main difference of the
simulation in the present study from that in Part 1 is the computational
domain size. In Part 1, the streamwise computational domain was $L_{x}/h=\pi$.
This size was chosen to match the most energetic streak instability length
scale of large-scale outer structures in the streamwise direction de
Giovanetti et al. (2017). However, the scope of the GQL in the present study
is to understand the scale interactions. For this purpose, the simulations
here are carried out by well resolving all the known coherent structures.
Since the largest streamwise and spanwise length scales in turbulent channel
flow are $\lambda_{x}/h\simeq 10$ and $\lambda_{z}/h\simeq 1$ del Alamo &
Jiménez (2003); Hwang (2015), a sufficiently large computational domain is
considered in the streamwise and spanwise directions: i.e. $L_{x}/h=8\pi$ and
$L_{z}/h=\pi$. We note that this domain size is 16 times bigger than that in
Part 1. Table 1 summarizes the parameters for the simulations performed in
this study. The Reynolds number for all the considered cases is
$Re=U_{0}h/\nu=66667$. The grid spacings are $\Delta_{x}^{+}=54.9-67.3$ and
$\Delta_{z}^{+}=27.5-33.6$. For each case, the number of spanwise Fourier
modes used in the GQL approximation is varied from $M_{z,F}=1$ (GQL1) to the
maximum number, the latter of which corresponds to that of the given LES.

Figure 3: Premultiplied spanwise wavenumber spectra of streamwise
$k_{z}^{+}\Phi_{uu}^{+}(y^{+},\lambda_{z}^{+})$ (left column) and wall-normal
$k_{z}^{+}\Phi_{vv}^{+}(y^{+},\lambda_{z}^{+})$ (right column) velocity for
(a,b) LES, (c,d) GQLZ1, (e,f) GQLZ5 and (g,h) GQLZ25 cases. Here, the vertical
line represents the spanwise cut-off wavelength ($\lambda_{z,c}$) dividing the
$\mathcal{P}_{h}$\- (left) and $\mathcal{P}_{l}$-subspace (right) regions.

##  3 GQL approximations

###  3.1 Turbulence statistics and spectra

The first- and second-order turbulence statistics as a function of the wall-
normal direction $y^{+}$ are plotted in figure 2 (the superscript
$(\cdot)^{+}$ denotes normalisation by viscous inner scale). There can be
observed good agreement between the LES and DNS statistics, plotted with
dashed line from Hoyas & Jiménez (2008) at $Re_{\tau}=2003$. Similarly to the
QL case in Part 1, the statistics of the GQLZ1 case are the most anisotropic
in comparison to the reference LES, although the physical nature of the flow
decomposition adopted in GQLZ1 is quite different from the QL case (see also
§4.1 and §4.2 for a further discussion). In particular, $u_{rms}^{+}$ (figure
2a) and $U^{+}$ (figure 2c) are larger than those of the reference LES,
behaving similarly to the QL model in the previous studies (Hernández et al.,
2021; Thomas et al., 2014; Farrell et al., 2016) where the quasilinear
approximation is made in the streamwise direction. The behaviour of
$v_{rms}^{+}$ (figure 2b) and $\langle
u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}$ depends on the $y$ location: they are
smaller on and below the log layer and greater above it. The enrichment of the
$\mathcal{P}_{l}$-subspace group by further incorporating more spanwise
Fourier modes (i.e. GQLZ5 and GQLZ25 cases) leads to the magnitude of the
near-wall streamwise velocity peak being considerably reduced, whilst
$v_{rms}^{+}$, $w_{rms}^{+}$ and $\langle
u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}$ become slightly larger than those of
the reference LES. For the GQLZ5 case, the inner-scaled mean velocity $U^{+}$
now provides a much better approximation to that of the reference LES and the
GQLZ1 case. As more spanwise Fourier modes are included in the
$\mathcal{P}_{l}$-subspace group, all the turbulence statistics converge to
those of the full LES (GQLZ5 and GQLZ25 cases in figure 2), similarly to the
GQL cases along the streamwise direction investigated in Part 1. Finally, the
statistics of the GQLZ25 case give the closest match to those of the reference
LES, as expected.

Figure 3 compares the premultiplied spanwise wavenumber spectra of streamwise
(left column) and wall-normal (right column) velocities of the reference LES
(figures 3a,b) with those of the GQLZ cases (figures 3c-h). Typical features
of the energy-containing motions in turbulent channel flow can be observed in
the spectra of LES (e.g. Hwang, 2015). The spanwise wavenumber spectra of
streamwise velocity appear to be aligned along a linear ridge $y\approx
0.1\lambda_{z}$ throughout the logarithmic region. At the bottom end of this
ridge ($y^{+}\approx 10$ and $\lambda_{z}^{+}\approx 100$), a local maximum
with the spanwise spacing of the near-wall streaks (Kline et al., 1967) is
found. At the top end ($y/h\approx 0.1$ and $\lambda_{z}/h\approx 1$), another
local maximum can be found, whose spanwise wavelength corresponds to the
spanwise length scale of large- and very-large-scale motions (Kovasznay et
al., 1970; del Alamo & Jiménez, 2003). The spanwise wavenumber spectra of
wall-normal velocity are also found to be aligned along a linear ridge
$y=0.5\lambda_{z}$ (figure 3a,b).

Figure 4: Premultiplied streamwise wavenumber spectra of streamwise
$k_{x}^{+}\Phi_{uu}^{+}(y^{+},\lambda_{x}^{+})$ (left column) and wall-normal
$k_{x}^{+}\Phi_{vv}^{+}(y^{+},\lambda_{x}^{+})$ (right column) velocity for
(a,b) LES, (c,d) GQLZ1, (e,f) GQLZ5 and (g,h) GQLZ25 cases.

The GQLZ1 case shows an increased energy in the region $\lambda_{z}^{+}\approx
500$ $(\lambda_{z}/h\approx 0.3)$ of the streamwise velocity spectra compared
to the LES case, in line with the statistics discussed above. This is in
contrast to the reduced energy featured at the other wavelengths (figure 3c).
The spanwise wavenumber spectra of wall-normal velocity show increased energy
in the region $\lambda_{z}^{+}\approx 1000$ $(\lambda_{z}/h\approx 0.5)$, and
the spectra span for most of the log region but showing little energy for
$\lambda_{z}^{+}\lesssim 200$ (figure 3d). The reduced energy observed for
$\lambda_{z}^{+}\lesssim 200$ explains the lack of Reynolds shear stress of
the GQLZ1 case in the near-wall region (figure 2d). By incorporating more
spanwise Fourier modes in the $\mathcal{P}_{l}$-subspace group (i.e. GQLZ5 and
GQLZ25), the peak location in the streamwise velocity spectra gradually moves
towards $(\lambda_{z}^{+},y^{+})\approx(100,10)$ along the linear ridge
$y=0.1\lambda_{z}$ and the outer part of the spectra also becomes more
energetic. The wall-normal velocity spectra now span the region close to the
wall along the linear ridge $y=0.5\lambda_{z}$. Moreover, the spectra of the
GQLZ25 case are found to be fairly similar to those of the reference LES. An
additional case GQLZ64 has been computed (see Appendix A) with
$\lambda_{z,c}^{+}\approx 97$, resulting in spanwise wavenumber spectra which
do not extend below $\lambda_{z,c}^{+}$, implying that the velocity field in
the $\mathcal{P}_{h}$-subspace group yields the trivial solution. This issue,
investigated in Part 1 for the streamwise GQL models, was explained in terms
of the Lyapunov exponent and the instability of the linearised equations in
the $\mathcal{P}_{h}$-subspace group.

The premultiplied streamwise wavenumber spectra of streamwise and wall-normal
velocities are shown in figure 4 for the LES and GQLZ cases. The energy-
containing motions in turbulent channel flow (LES) at a given spanwise length
scale are composed of a bimodal structure with two components: a long streaky
structure mainly carrying streamwise turbulent kinetic energy and a short and
tall vortex packet carrying turbulent kinetic energy at all the velocity
components (Hwang, 2015). The corresponding linear ridges $y\approx
0.01\lambda_{x}$ (lower line) and $y\approx 0.35\lambda_{x}$ (upper line),
along which these structures contribute to the spectra, are plotted in figure
4. The former corresponds to a long streaky structure and the streamwise
velocity spectra show high energy intensity along it (figure 4a). The latter
corresponds to a vortex packet and the streamwise spectra are less energetic
along it, whilst the wall-normal velocity spectra are very well aligned with
it. The streamwise velocity spectra of the GQLZ1 case are energetic in the
region $\lambda_{x}^{+}\gtrsim 600$, whilst the wall-normal velocity spectra
do not extend below $\lambda_{x}^{+}\lesssim 200$. The former appear to be
around a single point along $y\approx 0.01\lambda_{x}$, whilst weak
correlation with $y\approx 0.35\lambda_{x}$ is observed in the latter. As more
spanwise Fourier modes are further incorporated into the
$\mathcal{P}_{l}$-subspace group (figures 4e-h; i.e. GQLZ5 and GQLZ25), the
two spectra begin to show more energy at smaller $\lambda_{x}$ along the
linear ridges. As a consequence, in the GQLZ5 case, the near-wall region is
better resolved and the spectra reach out to smaller scales down to
$\lambda_{x}^{+}\approx 150$. The spectra of the GQLZ25 case greatly resemble
those of the LES case.

It is interesting to observe that the overall behaviours of the GQL models in
this study are similar to those of the streamwise QL and GQL models in Part 1
(e.g. Thomas et al., 2014, 2015; Farrell et al., 2016; Tobias & Marston, 2017;
Hernández & Hwang, 2020), although the flow decomposition in (1) implies
physically different nature of the GQL models here from those in Part 1. The
QL and GQL models in Part 1 are designed to offer a minimal description for
the self-sustaining processes at all integral length scales and study the
effect of their mutual nonlinear coupling: the full nonlinear equations in the
$\mathcal{P}_{l}$ subspace are primarily to describe the full nonlinear
evolution of the long streaks (or streamwise velocity fluctuations), whereas
the linearised equations in the $\mathcal{P}_{h}$ subspace are to resolve the
streak instability and the subsequent breakdown process of the streaks. On the
contrary, in the GQL models considered here, the full nonlinear equations in
the $\mathcal{P}_{l}$ subspace almost fully resolve every essential elements
of the self-sustaining process at least at the integral length scales for
$\lambda_{z}\geq\lambda_{z,c}$ ($\lambda_{z,c}=2\pi/k_{z,c}$), whereas the
linearised equations in the $\mathcal{P}_{h}$ subspace do not support its
presence for $\lambda_{z}<\lambda_{z,c}$. A further detailed discussion on
this issue will be given in §4.1 and §4.2.

###  3.2 Spectral energy transfer

Figure 5: Premultiplied spanwise wavenumber spectra of production
$k_{z}^{+}y^{+}\widehat{P}^{+}(y^{+},\lambda_{z}^{+})$ (left column) and
turbulent transport
$k_{z}^{+}y^{+}\widehat{T}_{turb}^{+}(y^{+},\lambda_{z}^{+})$ (right column)
for (a,b) LES, (c,d) GQLZ1, (e,f) GQLZ5 and (g,h) GQLZ25 cases. Here, the
vertical line represents the spanwise cut-off wavelength ($\lambda_{z,c}$)
dividing the $\mathcal{P}_{h}$\- (left) and $\mathcal{P}_{l}$-subspace (right)
regions.

Figure 6: Premultiplied streamwise wavenumber spectra of production
$k_{x}^{+}y^{+}\widehat{P}^{+}(y^{+},\lambda_{x}^{+})$ (left column) and
turbulent transport
$k_{x}^{+}y^{+}\widehat{T}_{turb}^{+}(y^{+},\lambda_{x}^{+})$ (right column)
for (a,b) LES, (c,d) GQLZ1, (e,f) GQLZ5 and (g,h) GQLZ25 cases.

In this section, the spectral energetics of the LES and GQLZ cases are further
considered. Given the present study is based an LES, the focus will be given
on the production and turbulent spectra only. Note that the pressure and
viscous transport spectra are negligibly small in the log and outer regions
(Cho et al., 2018; Lee & Moser, 2019). The premultiplied one-dimensional
spanwise wavenumber spectra of the production and turbulent transport of each
case are plotted in figure 5. The turbulent production spectra in figure 5(a)
appear to be almost uniformly distributed along the ridge $y\approx
0.2\lambda_{z}$ especially in the log layer. Owing to the energy conservative
nature of the nonlinear terms in the Navier-Stokes equations, the
premultiplied turbulent transport spectra in figure 5(b) show positive and
negative regions. The negative turbulent transport (blue contours in figure
5b) is almost balanced with the (positive) production in the logarithmic and
outer layers, and the positive turbulent transport appears along a ridge
indicating the Kolmogorov scale (i.e. $\lambda_{z}\approx 57\eta$, where
$\eta$ is the Kolmogorov scale). There can also be observed a region of weakly
positive turbulent transport spectra in the region very close to the wall
($y^{+}<10$) over a wide range of the spanwise wavelength scales
($200\delta_{\nu}\lesssim\lambda_{z}\lesssim 0.8h$), primarily due to an
inverse energy transfer from small to large scales Cho et al. (2018); Doohan
et al. (2021b).

The GQLZ1 case shows spectra with increased energy and a production peak
centred in $\lambda_{z}^{+}\approx 400$ (figures 5c,d). The production spectra
do not extend for $\lambda_{z}^{+}\lesssim 200$ and they are damaged at large
scales ($\lambda_{z}/h\gtrsim 1$), except in the outer region (figure 5c).
Moreover, similarly to the velocity spectra (figure 3), the linear scaling of
$\lambda_{z}^{+}$ with $y^{+}$ is found to have been weakened. The turbulent
transport spectra show positive and negative regions which have been reduced
over the entire spanwise wavelengths compared to the reference LES (figure
5d). Also, the weakly positive spectra in the region close to the wall is
confined to the largest spanwise scales $\lambda_{z}^{+}\approx 5000$. This is
somehow expected, because the linearised part of the GQLZ1 model would not
admit full energy cascade mechanism due to the absence of the self-interacting
nonlinear terms in the $\mathcal{P}_{h}$ subspace – the spanwise energy
cascade should only be admitted through the interaction between
$\mathbf{u}_{l}$ and $\mathbf{u}_{h}$ (i.e. the ‘scattering’ mechanism
proposed by Tobias & Marston, 2017). By enriching the
$\mathcal{P}_{l}$-subspace group with more spanwise Fourier modes, the
resulting spectra start to recover the original range of spanwise wavelengths,
the scaling with $y$ and the original position of the peak in the GQLZ5 and
GQLZ25 cases. The recovery of the spectra to those of the full LES, however,
appears to be in a little delicate manner, and this is seen in the turbulent
transport spectra of GQLZ25 (figure 5h). We note that turbulent transport
spectra for $\lambda_{z}<\lambda_{z,c}$ (i.e. the $\mathcal{P}_{h}$ subspace
group) in the outer region (say $y^{+}\gtrsim 300$) are almost zero, while
showing a sharp discontinuity across $\lambda_{z}=\lambda_{z,c}$. This
suggests that, in the outer region, the energy cascade from the
$\mathcal{P}_{l}$ to the $\mathcal{P}_{h}$ subspace does not exist in the
GQL25 model. This issue will be discussed further in §4.3.4.

The premultiplied streamwise wavenumber spectra of the energy budget are shown
in figure 6. For the full LES, the production spectra and the negative part of
the turbulent transport spectra appear to be in the region bounded by two
linear ridges, along which the wall-normal and streamwise velocity spectra are
highly energetic (i.e. $y=0.35\lambda_{x}$ and $y=0.01\lambda_{x}$ in figure
4). We note that, in the logarithmic region, the premultiplied production
spectra, $-k_{x}^{+}y^{+}\Phi_{uv}^{+}(k_{x},y)dU^{+}/dy^{+}$, are similar to
those of Reynolds shear-stress co-spectra, $-k_{x}^{+}\Phi_{uv}^{+}(k_{x},y)$,
due to $dU^{+}/dy^{+}\sim 1/y^{+}$ Cho et al. (2018). However, the linear
scaling of the production spectra with respect to $y$ is not very clear, and
this is presumably due to the relatively low Reynolds numbers in the present
study (for high Reynolds numbers, e.g. $Re_{\tau}\approx 5200$, see Lee &
Moser, 2019 and Hwang & Lee, 2020, where the linear scaling is clearly
visible). The turbulent transport spectra in figure 6(b) show a region of
positive values along the ridge corresponding to viscous dissipation
($\lambda_{x}=57\eta$) and a region of negative values corresponding to the
streamwise production intensity peak in the near-wall region
($\lambda_{x}^{+}\approx 1000$).

The spectra of GQLZ1 show some resemblance to the LES case but still a very
damaged energy cascade compared to that (figure 6c). In particular, the
production spectra exhibit an intense peak highly localised around
$\lambda_{x}^{+}\simeq 5000$ and $y^{+}\simeq 25$, while a secondary peak is
also observed in the outer region. The production spectra show no energy
intensity for wavelengths below $\lambda_{x}^{+}=700$ (figure 6c). The
positive region of the transport spectra appear at approximately the same
locations for larger streamwise length scales compared to LES (peaks at
$\lambda_{x}^{+}\approx 1000$ and $\lambda_{x}^{+}\approx 500$, respectively),
while the negative region is displaced to higher streamwise wavelengths and
reduced to the $y^{+}\approx 30$ and $y^{+}\approx 500$ locations. As more
spanwise modes are incorporated in the $\mathcal{P}_{l}$-subspace group, the
scalings of both spectra with $y$ and $\eta$ are greatly recovered, i.e. GQLZ5
and GQLZ25 cases (figures 6e-h).

###  3.3 Componentwise energy transport and pressure strain

Figure 7: Premultiplied spanwise wavenumber spectra of
$k_{z}^{+}y^{+}\widehat{\Pi}_{x}^{+}(y^{+},\lambda_{z}^{+})$ (left column) and
$k_{z}^{+}y^{+}\widehat{\Pi}_{yz}^{+}(y^{+},\lambda_{z}^{+})$ (right column)
for (a,b) LES, (c,d) GQLZ1, (e,f) GQLZ5 and (g,h) GQLZ25. Here, the vertical
line represents the spanwise cut-off wavelength ($\lambda_{z,c}$) dividing the
$\mathcal{P}_{h}$\- (left) and $\mathcal{P}_{l}$-subspace (right) regions. The
dashed lines indicate the ridge $y=0.5\lambda_{z}$.

Figure 8: Premultiplied streamwise wavenumber spectra of
$k_{x}^{+}y^{+}\widehat{\Pi}_{x}^{+}(y^{+},\lambda_{x}^{+})$ (left column) and
$k_{x}^{+}y^{+}\widehat{\Pi}_{yz}^{+}(y^{+},\lambda_{x}^{+})$ (right column)
for (a,b) LES, (c,d) GQLZ1, (e,f) GQLZ5 and (g,h) GQLZ25. The dashed lines
indicate the ridge $y=0.35\lambda_{x}$.

The pressure-strain spectra are finally explored to understand the mechanism
of componentwise TKE distribution in the GQL model. Figure 7 shows the
spanwise wavenumber spectra of the pressure-strain terms for LES and the GQL
models. For all cases, a negative $\widehat{\Pi}_{x}$ and mainly positive
$\widehat{\Pi}_{y}$ and $\widehat{\Pi}_{z}$ (combined into
$\widehat{\Pi}_{yz}=\widehat{\Pi}_{y}+\widehat{\Pi}_{z}$) can be observed in
the spanwise wavenumber space. This means that the pressure strain
redistributes the energy produced at the streamwise component of the TKE
through the production term to the wall-normal and spanwise components. The
spectra appear to be distributed approximately along the linear ridge
$y=0.2\lambda_{z}$. The GQLZ1 case underestimates the spectral energy
intensity of all pressure-strain terms, and their spanwise wavenumber spectra
do not extend to length scales below $\lambda_{z}^{+}\approx 200$. The two
features are improved as $\lambda_{z,c}$ is decreased, the spectra reach
smaller spanwise scales in the GQLZ5 case ($\lambda_{z}^{+}\approx 100$). Both
spectra of the $\mathcal{P}_{h}$ subspace group of GQLZ5 and GQLZ25 reproduce
with good agreement the spectra of the LES. Figure 8 shows the streamwise
wavenumber spectra of the pressure-strain terms. The pressure-strain intensity
appears to be concentrated for wavelengths $\lambda_{x}^{+}\gtrsim 300$ in the
GQLZ1 case, but the spectra quickly extend to smaller $\lambda_{x}$ as the
number of spanwise modes in the $\mathcal{P}_{l}$-subspace group is increased.
The streamwise spectra of GQLZ5 and GQLZ25 show excellent agreement with those
of the LES.

As was shown in Part 1, the GQL approximations only change the equations for
the ‘slow pressure’, $p_{l}^{S}$, given by

$\frac{1}{\rho}\nabla^{2}{p_{l}^{S}}=\mathcal{P}_{l}\Big{[}-\frac{\partial
u_{l,j}}{\partial x_{i}}\frac{\partial u_{l,i}}{\partial
x_{j}}\Big{]}+\mathcal{P}_{l}\Big{[}-2\frac{\partial u_{l,j}}{\partial
x_{i}}\frac{\partial u_{h,i}}{\partial
x_{j}}\Big{]}+\mathcal{P}_{l}\Big{[}-\frac{\partial u_{h,j}}{\partial
x_{i}}\frac{\partial u_{h,i}}{\partial x_{j}}\Big{]},$ (10a) in the
$\mathcal{P}_{l}$ subspace and
$\frac{1}{\rho}\nabla^{2}p_{h}^{S}=\mathcal{P}_{h}\Big{[}-\frac{\partial
u_{l,j}}{\partial x_{i}}\frac{\partial u_{l,i}}{\partial
x_{j}}\Big{]}+\mathcal{P}_{h}\Big{[}-2\frac{\partial u_{l,j}}{\partial
x_{i}}\frac{\partial u_{h,i}}{\partial
x_{j}}\Big{]}+\mathcal{P}_{h}\Big{[}-\frac{\partial u_{h,j}}{\partial
x_{i}}\frac{\partial u_{h,i}}{\partial x_{j}}\Big{]},$ (10b)

in the $\mathcal{P}_{h}$ subspace. In the GQL models, the second term in the
right-hand side of (10a) and the first and the last terms in the right-hand
side of (10b) are absent. Therefore, similarly to the GQL approximations
considered in the streamwise direction in Part 1, the GQL models here should
also experience the lack of pressure-strain transport mediated by such slow
pressure, explaining why the GQL models show highly anisotropic turbulent
fluctuations biased to the streamwise component especially when
$\lambda_{z,c}$ is large (see figure 2).

##  4 Discussion

###  4.1 Comparison with GQL approximations in the streamwise direction

Thus far, we have investigated a set of GQL approximations and compared its
turbulence statistics, spectra and energy transfer to those of full LES at
$Re_{\tau}\simeq 1700$ in turbulent channel flow. For the GQL approximations
in this study, the flow is decomposed into a group of low spanwise wavenumber
Fourier modes ($\mathcal{P}_{l}$ subspace) and the rest ($\mathcal{P}_{h}$
subspace): see (1). The former group is solved considering the full nonlinear
equations, whereas the latter is obtained by solving the linearised equations
about the former. The decomposition employed for the GQL models in the present
study evidently has different physical implications to that in Part 1. The QL
and GQL models in Part 1 (or RNL from Thomas et al., 2014, 2015; Farrell et
al., 2017) place a minimal model for the self-sustaining process at each of
the (spanwise) integral length scales and examine how their nonlinear coupling
dynamics and the related streamwise energy cascade play a role in the
description of full turbulent dynamics (i.e. the GQL models in Part 1). On the
contrary, in the present study, the flow field in the $\mathcal{P}_{l}$
subspace is supposed to fully incorporate the self-sustaining process at least
within the given integral length scales, while that in the $\mathcal{P}_{h}$
subspace is not expected to support the self-sustaining process (see §4.2).
Despite the significant difference in the physical implications between the
GQL approximations considered in Part 1 and in this study, it is worth
pointing out that the two different types of GQL models exhibit some notable
similarities, which would originate from the general nature of the GQL
approximation.

First, the GQL models with large cut-off wavelengths (i.e.
$\lambda_{x,c}(\equiv 2\pi/k_{x,c}$) in Part 1 and $\lambda_{z,c}$ in this
study) exhibit a considerably reduced multi-scale behaviour. Indeed, the QL
model in Part 1 and the GQL1 case in this study show spanwise wavenumber
spectra, where the energy density is focused on a particular length scale
(figures 3c,d in Part 1 and figures 3c,d) instead of being spread over a wide
range of the spanwise wavenumbers. As discussed in detail in §4.1 of Part 1,
the blockage of nonlinear energy transfer in the GQL approximations with a
large cut-off wavelength appears to elevate energy at a particular integral
length scale. This subsequently affects the mean-fluctuation dynamics in a way
that it does not strongly support turbulent fluctuations in the other wall-
normal locations and the corresponding spectral range, resulting in a reduced
multi-scale behaviour. It appears that this behaviour is rather common
regardless of the nature of the flow decomposition at least in the present
channel flow. This implies that a key to the successful prediction of low-
order turbulence statistics using the QL/GQL framework would lie in an
accurate description of the balanced mean-fluctuation interactions across the
wide range of integral length scales. It also explains why much simpler
quasilinear models Hwang & Eckhardt (2020); Skouloudis & Hwang (2021), the
focus of which was primarily given to modelling of the mean-fluctuation
interactions, were reasonably successful in the description of general
statistical behaviors of the same flow.

Second, when $\lambda_{z,c}$ or $\lambda_{x,c}$ is large, the GQL models in
Part 1 as well as in the present study exhibit a highly anisotropic turbulent
fluctuation which contains significantly elevated energy in the streamwise
component. Since the turbulence production in parallel wall-bounded shear
flows appears only in the streamwise component, this implies that, in general,
GQL approximations prevent the energy distribution mechanism into the other
velocity components. In fact, this is exactly the role played by the pressure
strain (e.g. Cho et al., 2018; Lee & Moser, 2019). Given the nature of any
QL/GQL approximations ignoring certain nonlinear fluctuation-fluctuation
interactions, it would always damage and/or disrupt the slow pressure
originating from the nonlinear fluctuation-fluctuation interactions,
explaining the highly anisotropic turbulent fluctuation of the QL/GQL models
in Part 1 and in the present study. This observation would be important if one
wishes to improve or develop a QL model simpler than the GQL models considered
here, as the role of the pressure-strain transport mediated by slow pressure
needs to be taken into account for the correct prediction of anisotropy of the
given flow.

###  4.2 Statistical description of near-wall turbulence and the self-
sustaining process

The near-wall turbulence approximately below $y^{+}\approx 50-70$ has been
understood to be largely independent of the motions in the logarithmic and
outer regions. It has consistently been shown that the absence and/or
disruption of the log- and outer-region motions in turbulence does not affect
the near-wall dynamics which attains the self-sustaining process at the
viscous inner length scale (e.g. Jiménez & Moin, 1991; Hamilton et al., 1995;
Jiménez & Pinelli, 1999; Hwang, 2013; Doohan et al., 2019). This early
observation extends to the logarithmic and outer regions, where similar self-
sustaining processes have been found to exist at larger scales in the form of
Townsend’s attached eddies Flores & Jiménez (2010); Hwang & Cossu (2010b,
2011); Hwang (2015); Hwang & Bengana (2016).

The self-sustaining process is a nonlinear mechanism that drives turbulence
utilising the energy from linearly stable mean flow $\mathbf{U}$. For a
mathematical description of the self-sustaining process in the
$\mathcal{P}_{h}$ subspace of a full simulation (or the full LES), let us
consider the full Navier-Stokes equations with $\mathbf{u}_{l}=\mathbf{0}$.
Employing the velocity decomposition used for the QL model in Part 1 (or the
RNL model), $\mathbf{u}_{h}$ is written as

$\mathbf{u}_{h}=\mathbf{u}_{h,0}+\mathbf{u}_{h,r},$ (11)

where $\mathbf{u}_{h,0}=\langle\mathbf{u}_{h}\rangle_{x}$. The equations for
$\mathbf{u}_{h,0}$ and $\mathbf{u}_{h,r}$ are then given by

$\displaystyle\frac{\partial\mathbb{u}_{h,0}}{\partial
t}+(\mathbb{U}\cdot\nabla)\textbf{u}_{h,0}+(\textbf{u}_{h,0}\cdot\nabla)\mathbb{U}=-\frac{1}{\rho}\nabla
p_{h,0}+\nu\mathbb{\nabla}^{2}\mathbb{u}_{h,0}$ (12a)
$\displaystyle-\mathcal{P}_{h}\left[\left(\textbf{u}_{h,0}\cdot\nabla\right)\textbf{u}_{h,0}\right]-\langle\mathcal{P}_{h}\left[\left(\textbf{u}_{h,r}\cdot\nabla\right)\textbf{u}_{h,r}\right]\rangle_{x},$
and $\displaystyle\frac{\partial\mathbb{u}_{h,r}}{\partial
t}+\left(\mathbb{U}\cdot\nabla\right)\textbf{u}_{h,r}+\left(\textbf{u}_{h,r}\cdot\nabla\right)\mathbb{U}=-\frac{1}{\rho}\nabla
p_{h,r}+\nu\mathbb{\nabla}^{2}\mathbb{u}_{h,r}$ (12b)
$\displaystyle-\mathcal{P}_{h}\left[\left(\textbf{u}_{h,0}\cdot\nabla\right)\mathbb{u}_{h,r}\right]-\mathcal{P}_{h}\left[\left(\mathbb{u}_{h,r}\cdot\nabla\right)\textbf{u}_{h,0}\right]$
$\displaystyle-\mathcal{P}_{h}\left[\left(\textbf{u}_{h,r}\cdot\nabla\right)\textbf{u}_{h,r}\right]+\langle\mathcal{P}_{h}\left[\left(\textbf{u}_{h,r}\cdot\nabla\right)\textbf{u}_{h,r}\right]\rangle_{x},$

where $p_{h,0}$ and $p_{h,r}$ are defined to enforce
$\mathbb{\nabla}\cdot\mathbb{u}_{h,0}=0$ and
$\mathbb{\nabla}\cdot\mathbb{u}_{h,r}=0$, respectively, with
$p_{h}=p_{h,0}+p_{h,r}$, and the eddy viscosity term for LES is ignored for
simplicity. Here, $\mathbf{u}_{h,0}$ in (12a) describes the evolution of the
streamwise elongated (or constant) streaks and rolls, whereas
$\mathbf{u}_{h,r}$ in (12b) depicts the streak instability (i.e. instability
of a streaky flow, $\mathbf{U}+\mathbf{u}_{h,0}$) – note that (12b) in the
absence of its third line containing self-interacting nonlinear terms gives
the linearised equations about $\mathbf{U}+\mathbf{u}_{h,0}$ in the
$\mathcal{P}_{h}$ subspace (i.e. the QL model in Part 1). From (12), it is
seen that $\mathbf{u}_{h,0}$ is driven by the presence of $\mathbf{u}_{h,r}$
(i.e. the last term in (12a)), whereas $\mathbf{u}_{h,r}$ physically
originates from the inflectional instability, transient growth and/or
parametric mechanism primarily depicted by the second term in the second line
of (12b) Hamilton et al. (1995); Schoppa & Hussain (2002); Park et al. (2011);
Farrell & Ioannou (2012); de Giovanetti et al. (2016, 2017); Lozano-Durán et
al. (2021). This is a two-way interaction between $\mathbf{u}_{h,0}$ (streaks)
and $\mathbf{u}_{h,r}$ (streamwise instability wave/quasi-streamwise
vortices), which sustains turbulence in the absence of $\mathbf{u}_{l}$.

Now, we consider the same form of the equations of motions for
$\mathbf{u}_{h}$ from the GQL models in the present study. Further to the
velocity decomposition in (11), $\mathbf{u}_{l}$ is also decomposed such that

$\mathbf{u}_{l}=\mathbf{u}_{l,0}+\mathbf{u}_{l,r},$ (13)

where $\mathbf{u}_{l,0}=\langle\mathbf{u}_{l}\rangle_{x}$. For the GQL models
in this study, the equations for $\mathbf{u}_{h,0}$ and $\mathbf{u}_{h,r}$ are
given by

$\displaystyle\frac{\partial\mathbb{u}_{h,0}}{\partial
t}+(\mathbb{U}\cdot\nabla)\textbf{u}_{h,0}+(\textbf{u}_{h,0}\cdot\nabla)\mathbb{U}=-\frac{1}{\rho}\nabla
p_{h,0}+\nu\mathbb{\nabla}^{2}\mathbb{u}_{h,0}$ (14a)
$\displaystyle-\mathcal{P}_{h}[\left(\textbf{u}_{h,0}\cdot\nabla\right)\textbf{u}_{l,0}]-\mathcal{P}_{h}[\left(\textbf{u}_{l,0}\cdot\nabla\right)\textbf{u}_{h,0}]$
$\displaystyle-\langle\mathcal{P}_{h}\left[\left(\textbf{u}_{h,r}\cdot\nabla\right)\textbf{u}_{l}\right]\rangle_{x}-\langle\mathcal{P}_{h}\left[\left(\textbf{u}_{l}\cdot\nabla\right)\textbf{u}_{h,r}\right]\rangle_{x},$
and $\displaystyle\frac{\partial\mathbb{u}_{h,r}}{\partial
t}+\left(\mathbb{U}\cdot\nabla\right)\textbf{u}_{h,r}+\left(\textbf{u}_{h,r}\cdot\nabla\right)\mathbb{U}=-\frac{1}{\rho}\nabla
p_{h,r}+\nu\mathbb{\nabla}^{2}\mathbb{u}_{h,r}$ (14b)
$\displaystyle-\mathcal{P}_{h}\left[\left(\textbf{u}_{h,r}\cdot\nabla\right)\mathbb{u}_{l}\right]-\mathcal{P}_{h}\left[\left(\mathbb{u}_{l}\cdot\nabla\right)\textbf{u}_{h,r}\right]$
$\displaystyle+\langle\mathcal{P}_{h}\left[\left(\textbf{u}_{h,r}\cdot\nabla\right)\textbf{u}_{l}\right]\rangle_{x}+\langle\mathcal{P}_{h}\left[\left(\textbf{u}_{l}\cdot\nabla\right)\textbf{u}_{h,r}\right]\rangle_{x}.$

Comparing (12) with (14), the differences between the turbulent fluctuations
from the self-sustaining process and the GQL models in the present study are
now evident. If (14a) admits a nontrivial $\mathbf{u}_{h,0}$ for $\mathbf{U}$
which does not yield any instability, it should be from the interaction
between $\mathbf{u}_{l}$ and $\mathbf{u}_{h}$ (the last two lines in (14a)).
In such a case, if $\mathbf{u}_{l}=\mathbf{0}$, there is no way for (14a) and
(14b) to have a non-trivial statistically stationary solution. Furthermore,
the term, which depicts an instability and/or transient growth mechanism from
a streaky flow (i.e. the second term in the second line of (12b)), does not
exist in (14b). In this case, an instability may possibly exist from the two
second terms respectively in the second and third lines of (14b), especially
when $\mathbf{u}_{l}$ is non-zero. Importantly, if $\mathbf{u}_{l}$ is a
chaotic velocity field, the non-trivial solution $\mathbf{u}_{h}$ in (14b)
emerges in the form of the leading neutrally stable Lyapunov vector of (14),
as was discussed in Farrell et al. (2017) and in Part 1.

Given the discussion above, it is crucial to have non-zero $\mathbf{u}_{l}$
for (14) (i.e. the equations in the $\mathcal{P}_{h}$ subspace in the present
study), whereas (12) does not need $\mathbf{u}_{l}$ to create its own
dynamics. Despite this important difference between turbulent fluctuations
from the self-sustaining process (12) and from the GQL model (14), it is
remarkable to observe that the GQL model admits velocity spectra fairly
similar to those of the full LES (GQLZ25, in particular; compare figures 4a,b
with 4g,h). We note that both (12) and (14) share the same Navier-Stokes
operator linearised about the mean velocity $\mathbf{U}$. In this respect,
they can also be viewed as the linearised Navier-Stokes operator driven by
chaotic fluctuations, the idea resonating with the previous and on-going
linearised Navier-Stokes-based modellling efforts for turbulent flows (e.g.
Hwang & Cossu, 2010a; McKeon & Sharma, 2010; Moarref et al., 2013; Zare et
al., 2017; McKeon, 2017; Hwang & Eckhardt, 2020; Skouloudis & Hwang, 2021;
Jovanović, 2021). Although such approaches would not be able to fully
incorporate some key nonlinear processes described in (12) and (14) without
additional modelling efforts, the observations here offer an explanation why
the linearised Navier-Stokes-based modelling efforts have been so useful to
study coherent structures in wall-bounded turbulence.

###  4.3 Scale interactions

0Case0 | 0(a)0 | 0(b)0 | 0(c)0 | 0(d)0 | 0(e)0 | 0(f)0
---|---|---|---|---|---|---
LES | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
GQL | ✓ | ✗ | ✓ | ✗ | ✓ | ✗
TRIAZ25 | ✓ | ✗ | ✓ | ✗ | ✓ | ✓
TRIBZ25 | ✓ | ✗ | ✗ | ✓ | ✓ | ✓
TRICZ25 | ✓ | ✗ | ✓ | ✓ | ✗ | ✓
TRIDZ25 | ✓ | ✗ | ✓ | ✓ | ✓ | ✗
Table 2: Triadic interactions in the present simulation cases. The tickmark
(✓) denotes inclusion whereas the crossmark (✗) stands for removal of the
corresponding triadic interaction. Here, the types of the triadic
interactions, indicated by alphabet, are described in figure 1.

((a)) $u_{rms}^{+}(y^{+})$

((b)) $v_{rms}^{+}(y^{+})$

((c)) $U^{+}(y^{+})$

((d)) $\langle u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}(y^{+})$

Figure 9: First- and second-order turbulence statistics for the present LES,
DNS at $Re_{\tau}=2003$ (Hoyas & Jiménez, 2008), TRIAZ25, TRIBZ25, TRICZ25 and
TRIDZ25 cases: (a) $u_{rms}^{+}(y^{+})$; (b) $v_{rms}^{+}(y^{+})$; (c)
$U^{+}(y^{+})$; (d) $\langle u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}(y^{+})$.

As discussed in §1, the GQL approximations can be utilised as an
interventional tool to examine scale interactions, given that it is capable of
suppressing particular triadic interactions in a controlled manner (see figure
1; see also table 2). Here, we consider the GQLZ25 case as a baseline, where
$\lambda_{z,c}^{+}\simeq 246$ ($\lambda_{z,c}/h\simeq 0.13$) is given for the
decomposition of velocity in (1). We subsequently carry out four additional
numerical experiments, in which certain types of triadic interactions between
$\mathcal{P}_{l}$\- and $\mathcal{P}_{h}$-subspace groups are artificially
suppressed. In principle, the same numerical experiments can repeatedly be
performed by considering different values of $\lambda_{z,c}$. However, in
general, this requires a large number of numerical experiments that depends on
the choice of $\lambda_{z,c}$, if one wishes to unveil the full triadic
interaction dynamics. This is beyond the scope of the present study.
Therefore, in the present study, we shall focus on the GQLZ25 case for the
purpose of conceptual demonstration.

The four additional numerical experiments are listed in table 2. Their first-
and the second-order turbulence statistics (mean and velocity fluctuations)
are reported in figure 9. The spanwise and streamwise wavenumber spectra of
streamwise and spanwise velocities are shown in figures 10 and 11, and those
of production and turbulent transport spectra are given in figures 12 and 13.

####  4.3.1 TRIAZ25: self-interactions within the $\mathcal{P}_{h}$ subspace

The TRIAZ25 case explores the contributions of the self-interacting nonlinear
terms in the $\mathcal{P}_{h}$ subspace to its own subspace (i.e.
$\mathcal{P}_{h}$ subspace), and this is represented by the triadic
interaction in figure 1(f). In the TRIAZ25 case, this interaction term is
activated further to the GQL25 model, and it has been designed to understand
the importance of such an interaction in the GQL approximation. In fact,
figure 9 shows that the statistics of TRIAZ25 resembles that of GQLZ25 the
most, compared to other cases, showing that the effect of this triadic
interaction on the low-order statistics of the GQL approximation is moderate.
The spanwise wavenumber velocity spectra of the TRIAZ25 case (figures 10a,b
and 11a,b) are also almost identical to those of the GQLZ25 case (figures 3g,h
and 4g,h). The same holds true for most of the energy spectra (figures 12a,b
and 13a,b), although some minor differences can be observed in the near-wall
region.

It should be mentioned that the addition of the triadic interaction in figure
1(f) is directly associated with the activation of the nonlinear terms
associated with the self-sustaining process in the $\mathcal{P}_{h}$ subspace.
This becomes evident from (12), where the contributions of the self-
interacting nonlinear terms in the $\mathcal{P}_{h}$ subspace to its own
subspace (i.e. $\mathcal{P}_{h}$ subspace) were shown to be directly
associated with some key sub-processes of the self-sustaining process. A
possible explanation to this is that there may exist competition and/or
cooperation between the self-sustaining process and the scattering mechanism
in driving the near-wall turbulent fluctuations, and in the presence of all
scales of turbulent flow, the scattering mechanism may also play an important
role.

Figure 10: Premultiplied spanwise wavenumber spectra of streamwise
$k_{z}^{+}\Phi_{uu}^{+}(y^{+},\lambda_{z}^{+})$ (left column) and wall-normal
$k_{z}^{+}\Phi_{vv}^{+}(y^{+},\lambda_{z}^{+})$ (right column) velocity for
(a,b) TRIAZ25, (c,d) TRIBZ25, (e,f) TRICZ25 and (g,h) TRIDZ25 cases.

Figure 11: Premultiplied streamwise wavenumber spectra of streamwise
$k_{x}^{+}\Phi_{uu}^{+}(y^{+},\lambda_{x}^{+})$ (left column) and wall-normal
$k_{x}^{+}\Phi_{vv}^{+}(y^{+},\lambda_{x}^{+})$ (right column) velocity for
(a,b) TRIAZ25, (c,d) TRIBZ25, (e,f) TRICZ25 and (g,h) TRIDZ25 cases. Here, the
vertical line represents the spanwise cut-off wavelength ($\lambda_{z,c}$)
dividing the $\mathcal{P}_{h}$\- (left) and $\mathcal{P}_{l}$-subspace (right)
regions.

####  4.3.2 TRIBZ25: inverse energy transfer

The TRIBZ25 case eliminates the contributions of the self-interacting
nonlinear terms in the $\mathcal{P}_{h}$ subspace to the
$\mathcal{P}_{l}$-subspace group (i.e. the inverse energy transfer from the
$\mathcal{P}_{h}$ to the $\mathcal{P}_{l}$ subspace), but activates all
triadic interactions in the $\mathcal{P}_{h}$-subspace group. In this case, it
is found that its turbulence statistics shows major differences with respect
to GQLZ25 and the other cases (figure 9). First, $-\langle
u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}$ is greater than one, and
$u_{rms}^{+}$ and $v_{rms}^{+}$ are also much greater than those of GQLZ25.
This strong overestimation of turbulent fluctuations of TRIBZ25 can be
explained with the following dimensionless streamwise mean-momentum equation:

$\frac{dU^{+}}{dy^{+}}-\frac{\langle
u_{l}v_{l}\rangle_{x,z,t}}{u_{\tau}^{2}}-\frac{\langle
u_{h}v_{h}\rangle_{x,z,t}}{u_{\tau}^{2}}=1-\frac{y}{h}.$ (15)

In the TRIBZ25 case, the third term in (15) is artificially omitted, thus the
mean is only balanced with $\langle u_{l}v_{l}\rangle_{x,z,t}$. We note that
the present simulation imposes a constant mass flow rate and that $\langle
u_{h}v_{h}\rangle_{x,z,t}$ is not into account in (15) in the TRIBZ25 case.
Therefore, no matter how strong $-{\langle u_{h}v_{h}\rangle_{x,z,t}}$ is, the
mean shear has no direct control in reducing its strength, explaining the
breakdown of (15) in the TRIBZ25 case.

Figure 12: Premultiplied spanwise wavenumber spectra of production
$k_{z}^{+}y^{+}\widehat{P}^{+}(y^{+},\lambda_{z}^{+})$ (left column) and
turbulent transport
$k_{z}^{+}y^{+}\widehat{T}_{turb}^{+}(y^{+},\lambda_{z}^{+})$ (right column)
for (a,b) TRIAZ25, (c,d) TRIBZ25, (e,f) TRICZ25 and (g,h) TRIDZ25 cases.

Figure 13: Premultiplied streamwise wavenumber spectra of production
$k_{x}^{+}y^{+}\widehat{P}^{+}(y^{+},\lambda_{x}^{+})$ (left column) and
turbulent transport
$k_{x}^{+}y^{+}\widehat{T}_{turb}^{+}(y^{+},\lambda_{x}^{+})$ (right column)
for (a,b) TRIAZ25, (c,d) TRIBZ25, (e,f) TRICZ25 and (g,h) TRIDZ25 cases. Here,
the vertical line represents the spanwise cut-off wavelength ($\lambda_{z,c}$)
dividing the $\mathcal{P}_{h}$\- (left) and $\mathcal{P}_{l}$-subspace (right)
regions.

In fact, the spanwise wavenumber spectra of velocity and energy budget in
figures 10(c,d) and 12(c,d) show large elevation of the fluctuation energy in
the $\mathcal{P}_{h}$ subspace in the TRIBZ25 case. This effect is also seen
in the streamwise wavenumber spectra in figures 11(c,d) and 13(c,d), where
spectral intensity is quite high in the region close to the wall where the
spectral intensity of $\mathcal{P}_{h}$-subspace group would be important.
This behaviour might have given an impression that the range of spanwise
wavenumbers covered by the spectra is reduced in the TRIBZ25 case. However, it
is important to note that turbulent velocity fluctuations in the outer regions
remain more or less the same as those of GQLZ25 and LES (see figure 9),
indicating that the spectral intensity of the outer region in the
$\mathcal{P}_{l}$ subspace is actually not affected significantly. The same
behaviour is also observed in all the other spectra in figures 11(c,d),
12(c,d) and 13(c,d) (not shown). This confirms the outer-similarity hypothesis
of Townsend (1976), who conjectured that turbulence in the outer region
remains largely independent of the near-wall dynamics, consistent with many
previous studies conducted with surface roughness (e.g. Flores et al., 2007).

Finally, it should be mentioned that the spanwise wavenumber spectra of
turbulent transport in the TRIBZ25 case exhibit a strong relative suppression
of the near-wall positive turbulent transport in the $\mathcal{P}_{h}$
subspace (figure 12d) in comparison to LES (figure 5b) and GQLZ25 (figure 5h).
The weakly positive turbulent transport observed in GQL25 and L£S was
originally proposed to be the main consequence of inverse energy cascade (Cho
et al., 2018), who visualised the full triadic nonlinear interactions in a
turbulent channel flow. However, some recent studies have suggested that it is
also generated by the self-sustaining process at each scale (Lee & Moser,
2019; Kawata & Tsukahara, 2021). The TRIBZ25 simulation here shows that the
positive turbulent transport in the region close to the wall is significantly
weakened by the inhibition of the energy transfer from the $\mathcal{P}_{h}$
to the $\mathcal{P}_{l}$ subspace, favouring the proposition by Cho et al.
(2018) on its primary origin.

####  4.3.3 TRICZ25: the scattering mechanism in GQL approximations

The TRICZ25 case enables the contributions of the self-interacting nonlinear
terms in both the $\mathcal{P}_{l}$ and $\mathcal{P}_{h}$ subspaces to the
$\mathcal{P}_{h}$-subspace group (previously eliminated by the GQL
approximation), while it excludes the low/high wavenumber interaction terms in
the $\mathcal{P}_{h}$-subspace group, i.e. the ‘scattering mechanism’ in the
GQL models (e.g. Tobias & Marston, 2017). The statistics of TRICZ25 shows some
subtle differences from that of GQLZ25 (see figure 9). In particular,
$u_{rms}^{+}$ (figure 9a) appears to be larger, while $\langle
u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}$ depends on the $y$-location: they are
greater on the log layer and smaller below and above it.

The spectra of the TRICZ25 case show more precise picture in the subtle
differences (figures 10e,f, 11e,f, 12e,f and 13e,f). In particular, the
spanwise wavenumber spectra of velocity and energy budget in figures 10(e,f)
and 12(e,f) are found to extend over a narrower range of spanwise wavenumbers.
Furthermore, the spectral intensity in the $\mathcal{P}_{h}$-subspace (for
$\lambda_{z}<\lambda_{z,c}$) is significantly weakened, not covering
wavenumbers below $\lambda_{z}^{+}\approx 100$. In particular, the spanwise
wavenumber spectra of turbulence production is completely suppressed (figure
12e). This suggests that the low/high wavenumber interaction terms in the
$\mathcal{P}_{h}$-subspace group, incorporated in the GQL approximations, are
probably the most crucial triadic interactions, because their suppression
significantly weakens the energy-containing region of the spanwise wavenumber
spectra of velocity.

It is interesting to note that this observation favours that the scattering
mechanism is more important than the self-sustaining process for the
generation of near-wall turbulence, when all the other scales are present in
the flow – we note that all the previous numerical experiments demonstrating
the self-sustaining process was without the motions larger than the given
spanwise length scale (e.g. Jiménez & Moin, 1991; Hamilton et al., 1995;
Hwang, 2013). However, care needs to be taken in understanding this behaviour,
because the TRICZ25 case also suppresses all the production terms in (14)
(i.e. the third terms in (14a) and (14b)), shared by the self-sustaining
process (see (12)). In this respect, the issue on the relative importance
between the self-sustaining process and the scattering mechanism in the
generation of near-wall turbulence still remains an open question, although
the numerical experiment carried out with the TRIAZ25 case (e.g. see figure
10a,b) appears to support that perhaps the scattering mechanism may be more
crucial than the self-sustaining process in the presence of all the scales.
Since the existence of the self-sustaining processes and their roles have been
well established Jiménez & Moin (1991); Hamilton et al. (1995); Hwang (2013),
a more careful examination with additional numerical experiments would
certainly be required for the precise understanding on this new observation,
and we will leave this for the future investigation.

####  4.3.4 TRIDZ25: direct driving from the $\mathcal{P}_{l}$ to the
$\mathcal{P}_{h}$ subspaces

Finally, the TRIDZ25 case incorporates the contribution of the self-
interacting nonlinear term in the $\mathcal{P}_{l}$ subspace to the
$\mathcal{P}_{h}$-subspace group, and its objective is to analyse the
influence of this direct driving mechanism from $\mathcal{P}_{l}$\- to
$\mathcal{P}_{h}$-subspace group. The statistics of TRIDZ25 is found to
approximate that of GQLZ25 fairly well (figure 9), with a little
overestimation of $u_{rms}^{+}$, $v_{rms}^{+}$, $\langle
u^{\prime}v^{\prime}\rangle_{x,z,t}^{+}$, showing that the inclusion of the
triadic interaction (d) has a moderate effect on the statistics.

The spectra of the TRIDZ25 case offer a more detailed picture on its effect
(figures 10g,h, 11g,h, 12g,h and 13g,h). The spanwise wavenumber velocity
spectra in figures 10(g,h) appear to be qualitatively similar to those of the
GQLZ25 case. However, the streamwise and wall-normal velocity spectra in the
$\mathcal{P}_{h}$ subspace (for $\lambda_{z}<\lambda_{z,c}$) are intensified
compared to those of GQLZ25 case (figures 3g,h), especially with a large
elevation of the energy in the relatively outer region. This can be explained
by the spanwise wavenumber spectra of turbulent transport in figure 12(h),
where a strong positive energy transport now appears in the outer region of
$\mathcal{P}_{h}$ subspace. This is in contrast to that of the GQLZ25 case
(figure 5h), where such a transport barely exist in the relative outer region
of the $\mathcal{P}_{h}$ subspace (i.e. $y^{+}\gtrsim 300$).

In Part 1, it was proposed that the $\mathcal{P}_{h}$ subspace group of a GQL
model can admit the trivial solution, if the cut-off wavelength for the
decomposition of the flow (i.e. $\lambda_{z,c}$) is sufficiently small (see
§4.2 in Hernández et al., 2021). This was also explained in terms of the
Lyapunov exponent of the $\mathcal{P}_{h}$ subspace group, which can
alternatively be interpreted as the activation process of the scattering
mechanism proposed by Tobias & Marston (2017). In the present study, such a
case also appears for $\lambda_{z,c}^{+}\approx 97$ (GQLZ64 in table 1 and in
Appendix A), the typical spanwise spacing of the near-wall streaks Kline et
al. (1967). In this case, the simulation result of the TRIDZ25 case suggests
that the contribution of the self-interacting nonlinear term in the
$\mathcal{P}_{l}$ subspace to the $\mathcal{P}_{h}$-subspace group becomes
important, when the scattering mechanism discussed in §4.3.3 plays little role
in generating the fluctuation in the $\mathcal{P}_{h}$ subspace (i.e. when
$\lambda_{z,c}$ is sufficiently small). In such a case, the dynamics in the
$\mathcal{P}_{h}$ subspace remains essentially passive to the
$\mathcal{P}_{l}$ subspace in the sense that the $\mathcal{P}_{h}$ subspace
would only admit a non-trivial solution in the presence of the direct driving
from the $\mathcal{P}_{l}$ subspace. In this respect, it is finally worth
mentioning that the smallest spanwise computational domain retaining
turbulence is $L_{z}^{+}\approx 100$ Jiménez & Moin (1991); Hwang (2013) and
the critical spanwise wavelength, below which the GQL model in this study
yields the trivial solution in the $\mathcal{P}_{h}$ subspace, is
$\lambda_{z,c}^{+}\approx 97$, showing an interesting analogue.

##  5 Concluding remarks

In the present study, a generalised quasilinear (GQL) approximation has been
applied to in turbulent channel flow at $Re_{\tau}\approx 1700$, continuing
from the work of Part 1 (Hernández et al., 2021). The flow is decomposed into
two groups, the former of which contains a set of low-wavenumber spanwise
Fourier modes and the latter are composed of the rest high-wavenumber modes.
The former group is then solved by considering the full nonlinear equations,
while the latter group is obtained from the linearised equations about the
former. In Part 1, the flow decomposition was based on the streamwise Fourier
modes. By doing so, the corresponding GQL models placed a minimal model for
self-sustaining process at all integral length scales varying from viscous
inner units to outer units, and the nonlinear coupling between the self-
sustaining processes from different scales and the related nonlinear energy
transfer were assessed. However, in this study, the flow decomposition gives
the low wavenumber group in which the self-sustaining process are fully
resolved at least within the given integral length scales, while the
linearised high wavenumber group is not able to support the self-sustaining
process. Some key conclusions are summarised as follows:

  1. 1.

Despite the important physical difference of the GQL approximation here from
that in Part 1, they share some important similarities. First, due to the
interference of nonlinear energy transfer given by GQL approximations, they
tend to exhibit significantly reduced multi-scale behaviours especially when
the cut-off wavelength for the flow decomposition is sufficiently large.
Furthermore, turbulent fluctuations have been found to be strongly anisotropic
for large cut-off wavelengths, and this is due to damaged slow pressure
originating from the fluctuation/fluctuation interactions, some of which are
omitted in the GQL approximations.

  2. 2.

While the high wavenumber group of the GQL models in the present study cannot
support the self-sustaining process, they can generate some key statistical
features at integral scales through the scattering mechanism to certain
extent. Although it is not clear how this is exactly possible, this
observation could have some potential links to the success of the previous
linearised Navier-Stokes operator-based models (e.g. Hwang & Cossu, 2010a;
McKeon & Sharma, 2010; Moarref et al., 2013; Zare et al., 2017; McKeon, 2017;
Hwang & Eckhardt, 2020; Skouloudis & Hwang, 2021; Jovanović, 2021).

  3. 3.

Finally, a set of numerical experiments have been performed to explore the
roles of ignored triadic interaction terms in the GQL approximations focusing
on scale interactions. These numerical experiments have clarified the role of
1) self-interactions within the high-wavenumber group; 2) inverse energy
transfer in the region close to the wall; 3) the scattering mechanism; 4)
direct driving mechanism to the the high-wavenumber group. In particular, it
has been seen that the scattering mechanism may play a more important role
than the self-sustaining process in driving near-wall turbulence in the
presence of all scales in the flow.

The GQL approximations were originally introduced to improve simpler QL
models. Here, we have utilised the GQL approximations to explore the coupling
dynamics between self-sustaining processes at different length scales, scale
interactions and nonlinear energy transfer for wall-bounded turbulence. The
knowledge gained here would be useful to develop a simpler and more robust
self-consistent description of turbulence using the concepts of the
‘statistical-state dynamics’. Our on-going efforts have been made along this
direction, and some recent works can be found in Hwang & Eckhardt (2020) and
Skouloudis & Hwang (2021).

## Acknowledgements

C. G. H. and Y. H. acknowledge the support of the Leverhulme trust
(RPG-123-2019). Q. Y. acknowledges the Natural Science Foundation of China
(11802322). Y. H. is also supported by the Engineering and Physical Sciences
Research Council (EPSRC; EP/T009365/1) in the UK.

## Declaration of interest

The authors report no conflict of interest.

##  Appendix A Velocity and energy-balance spectra of GQLZ64

Figure 14: Premultiplied spanwise wavenumber spectra of (a) streamwise
$k_{z}^{+}\Phi_{uu}^{+}(y^{+},\lambda_{z}^{+})$ and (b) wall-normal
$k_{z}^{+}\Phi_{vv}^{+}(y^{+},\lambda_{z}^{+})$ velocity, (c) production
$k_{z}^{+}y^{+}\widehat{P}^{+}(y^{+},\lambda_{z}^{+})$ and (d) turbulent
transport $k_{z}^{+}y^{+}\widehat{T}_{turb}^{+}(y^{+},\lambda_{z}^{+})$ for
GQLZ64. Here, the vertical line represents the spanwise cut-off wavelength
($\lambda_{z,c}$) dividing the $\mathcal{P}_{h}$\- (left) and
$\mathcal{P}_{l}$-subspace (right) regions.

Figure 14 shows the premultiplied spanwise wavenumber spectra of the
streamwise velocity (figure 14a), the wall-normal velocity (figure 14b), the
production (figure 14c) and the turbulent transport (figure 14d) of the GQLZ64
case. All spectra greatly resemble those of the reference LES in the
$\mathcal{P}_{l}$ subspace. The trivial solution $\mathbf{u}_{h}=0$ is however
observed in the $\mathcal{P}_{h}$-subspace group, for a cutoff wavelength
$\lambda_{z,c}^{+}\approx 97$. This phenomenon, investigated in Part 1 for the
streamwise GQL models, has been explained in terms of the Lyapunov exponent
and the instability of the linearised equations in the
$\mathcal{P}_{h}$-subspace group (Hernández et al., 2021).

## References

  * Agostini & Leschziner (2016) Agostini, L. & Leschziner, M. 2016 Predicting the response of small-scale near-wall turbulence to large-scale outer motions. Phys. Fluids 28 (1), 015107.
  * del Alamo & Jiménez (2003) del Alamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41–L44.
  * Cho et al. (2018) Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. J. Fluid Mech. 854, 474–504.
  * Constantinou (2015) Constantinou, N. C. 2015 Formation of large-scale structures by turbulence in rotating planets. Ph.D. thesis, National and Kapodistrian University of Athens, Athens .
  * Doohan et al. (2021a) Doohan, P., Bengana, Y., Yang, Q., Willis, A. P. & Hwang, Y. 2021a Multi-scale state space and travelling waves in wall-bounded turbulence Under review.
  * Doohan et al. (2019) Doohan, P., Willis, A. P. & Hwang, Y. 2019 Shear stress-driven flow: the state space of near-wall turbulence at infinite friction Reynolds number. J. Fluid Mech. 874, 606–638.
  * Doohan et al. (2021b) Doohan, P., Willis, A. P. & Hwang, Y. 2021b Minimal multi-scale dynamics of near-wall turbulence. J. Fluid Mech. 913, A8.
  * Duvvuri & McKeon (2015) Duvvuri, S. & McKeon, B. J. 2015 Triadic scale interactions in a turbulent boundary layer. J. Fluid Mech. 767, R4.
  * Farrell & Ioannou (2012) Farrell, B.F. & Ioannou, P.J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.
  * Farrell et al. (2016) Farrell, B.F., Ioannou, P.J., Jiménez, J., Constantinou, N.C., Lozano-Durán, A. & Nikolaidis, M-A 2016 A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809, 290–315.
  * Farrell et al. (2017) Farrell, B. F., Gayme, D. F. & Ioannou, P. J. 2017 A statistical state dynamics approach to wall turbulence. Phil. Trans. R. Soc. A pp. 375, 20160081.
  * Farrell & Ioannou (2003) Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118.
  * Farrell & Ioannou (2007) Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64 (10), 3652–3665.
  * Flores & Jiménez (2010) Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22 (7), 071704\.
  * Flores et al. (2007) Flores, O., Jiménez, J. & Del Álamo, J. C. 2007 Vorticity organization in the outer layer of turbulent channels with disturbed walls. J. Fluid Mech. 591, 145–154.
  * de Giovanetti et al. (2016) de Giovanetti, M., Hwang, Y. & Choi, H. 2016 Skin-friction generation by attached eddies in turbulent channel flow. J. Fluid Mech. 808, 511–538.
  * de Giovanetti et al. (2017) de Giovanetti, M., Sung, H. J. & Hwang, Y. 2017 Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions. J. Fluid Mech. 832, 483–513.
  * Hamilton et al. (1995) Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.
  * Hernández & Hwang (2020) Hernández, C. G. & Hwang, Y. 2020 Spectral energetics of a quasilinear approximation in uniform shear turbulence. J. Fluid Mech. 904, A11.
  * Hernández et al. (2021) Hernández, C. G., Yang, Q. & Hwang, Y. 2021 Generalised quasilinear approximations of turbulent channel flow: Part 1. streamwise nonlinear energy transfer arXiv:2108.12395.
  * Herring (1963) Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 (4), 325–338.
  * Herring (1964) Herring, J. R. 1964 Investigation of problems in thermal convection: Rigid boundaries. J. Atmos. Sci. 21 (3), 277–290.
  * Herring (1966) Herring, J. R. 1966 Some analytic results in the theory of thermal convection. J. Atmos. Sci. 23 (6), 672–677.
  * Hoyas & Jiménez (2008) Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20 (10), 101511.
  * Hutchins & Marusic (2007) Hutchins, N. & Marusic, I. 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. A:Math. Phys. Engng Sci. 365 (1852), 647–664.
  * Hwang (2013) Hwang, Y. 2013 Near-wall turbulent fluctuations in the absence of wide outer motions. J. Fluid Mech. 723, 264–288.
  * Hwang (2015) Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254–289.
  * Hwang & Bengana (2016) Hwang, Y. & Bengana, Y. 2016 Self-sustaining process of minimal attached eddies in turbulent channel flow. J. Fluid Mech. 795, 708–738.
  * Hwang & Cossu (2010a) Hwang, Y. & Cossu, C. 2010a Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.
  * Hwang & Cossu (2010b) Hwang, Y. & Cossu, C. 2010b Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett. 105, 044505.
  * Hwang & Cossu (2011) Hwang, Y. & Cossu, C. 2011 Self-sustained processes in the logarithmic layer of turbulent channel flows. Phys. Fluids 23 (6), 061702.
  * Hwang & Eckhardt (2020) Hwang, Y. & Eckhardt, B. 2020 Attached eddy model revisited using a minimal quasi-linear approximation. J. Fluid Mech. 894, A23.
  * Hwang & Lee (2020) Hwang, Y. & Lee, M. 2020 The mean logarithm emerges with self-similar energy balance. J. Fluid Mech. 903, R6.
  * Jiménez & Moin (1991) Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.
  * Jiménez & Pinelli (1999) Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.
  * Jovanović (2021) Jovanović, M. R. 2021 From bypass transition to flow control and data-driven turbulence modeling: An input–output viewpoint. Ann. Rev. Fluid Mech. 53 (1), 311–345.
  * 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.
  * Kawata & Tsukahara (2021) Kawata, T. & Tsukahara, T. 2021 Scale interactions in turbulent plane couette flows in minimal domains. J. Fluid Mech. 911, A55.
  * Kline et al. (1967) Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30 (4), 741–773.
  * Kovasznay et al. (1970) Kovasznay, L. S. G., Kibens, V. & Blackwelder, R. F. 1970 Large-scale motion in the intermittent region of a turbulent boundary layer. J. Fluid Mech. 41 (2), 283–325.
  * Lee & Moser (2019) Lee, M. K. & Moser, R. D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886–938.
  * Lozano-Durán et al. (2021) Lozano-Durán, A., Constantinou, N. C., Nikolaidis, M.-A. & Karp, M. 2021 Cause-and-effect of linear mechanisms sustaining wall turbulence. J. Fluid Mech. 914, A8.
  * Malkus (1954) Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Phil. Trans. R. Soc. A 225 (1161), 196–212.
  * Malkus (1956) Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1 (5), 521–539.
  * Mantič-Lugo et al. (2014) Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113, 084501.
  * Mantič-Lugo & Gallaire (2016) Mantič-Lugo, V. & Gallaire, F. 2016 Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation. Phys. Rev. Fluids 1, 083602.
  * Marston et al. (2016) Marston, J. B., Chini, G. P. & Tobias, S. M. 2016 Generalized quasilinear approximation: Application to zonal jets. Phys. Rev. Lett. 116, 214501.
  * Marston et al. (2008) Marston, J. B., Conover, E. & Tobias, S. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65 (6), 1955–1966.
  * Mathis et al. (2009) Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.
  * McKeon (2017) McKeon, B. J. 2017 The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817, P1.
  * McKeon & Sharma (2010) McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.
  * Mizuno (2016) Mizuno, Y. 2016 Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. J. Fluid Mech. 805, 171–187.
  * Moarref et al. (2013) Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling of the streamwise energy density in high-reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.
  * Park et al. (2011) Park, J., Hwang, Y. & Cossu, C. 2011 On the stability of large-scale streaks in turbulent Couette and Poiseulle flows. C. R. Mécanique 339 (1), 1 – 5.
  * Pausch et al. (2019) Pausch, M., Yang, Q., Hwang, Y. & Eckhardt, B. 2019 Quasilinear approximation for exact coherent states in parallel shear flows. Fluid Dyn. Res. 51 (1), 011402.
  * Schoppa & Hussain (2002) Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.
  * Skouloudis & Hwang (2021) Skouloudis, N. & Hwang, Y. 2021 Scaling of turbulence intensities up to ${Re}_{\tau}={10}^{6}$ with a resolvent-based quasilinear approximation. Phys. Rev. Fluids 6, 034602.
  * Thomas et al. (2015) Thomas, V. L., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27 (10), 105104.
  * Thomas et al. (2014) Thomas, V. L., Lieu, B. K., Jovanović, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26 (10), 105112.
  * Tobias & Marston (2013) Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110, 104502.
  * Tobias & Marston (2017) Tobias, S. M. & Marston, J. B. 2017 Three-dimensional rotating Couette flow via the generalised quasilinear approximation. J. Fluid Mech. 810, 412–428.
  * Townsend (1976) Townsend, A. 1976 The Structure of Turbulent Shear Flow. 2nd edn. Cambridge University Press.
  * Vreman (2004) Vreman, A. W. 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications. Phys. Fluids 16 (10), 3670–3681.
  * Waleffe (1997) Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.
  * Zare et al. (2017) Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.
  * Zhang & Chernyshenko (2016) Zhang, C. & Chernyshenko, S. I. 2016 Quasisteady quasihomogeneous description of the scale interactions in near-wall turbulence. Phys. Rev. Fluids 1, 014401.