# Spectral analysis on transport budgets of turbulent heat fluxes in plane
Couette turbulence

T. Kawata  Keio University, Hiyoshi 3-14-1, Yokohama, 223-8522 Kanagawa, Japan
<EMAIL_ADDRESS>T. Tsukahara  Tokyo University of Science, Yamazaki 2641, Noda,
278-8510 Chiba, Japan

###### Abstract

In recent years, scale-by-scale energy transport in wall turbulence has been
intensively studied, and the complex spatial and interscale transfer of
turbulent energy has been investigated. As the enhancement of heat transfer is
one of the most important aspects of turbulence from an engineering
perspective, it is also important to study how turbulent heat fluxes are
transported in space and in scale by nonlinear multi-scale interactions in
wall turbulence as well as turbulent energy. In the present study, the
spectral transport budgets of turbulent heat fluxes are investigated based on
direct numerical simulation data of a turbulent plane Couette flow with a
passive scalar heat transfer. The transport budgets of spanwise spectra of
temperature fluctuation and velocity-temperature correlations are investigated
in detail in comparison to those of the corresponding Reynolds stress spectra.
The similarity and difference between those scale-by-scale transports are
discussed, with a particular focus on the roles of interscale transport and
spatial turbulent diffusion. As a result, it is found that the spectral
transport of the temperature-related statistics is quite similar to those of
the Reynolds stresses, and in particular inverse interscale transfer is
commonly observed throughout the channel in both transport of the Reynolds
shear stress and wall-normal turbulent heat flux.

##  I  Introduction

Analysing the budgets of the Reynolds stress transport equation is a useful
tool to investigate turbulence transport phenomena, as it shows quantitatively
where and how much turbulence is produced from the mean flow, redistributed
among different velocity components by the effect of pressure, dissipated into
heat by viscosity, etc. A particularly interesting feature is that the budget
includes spatial transport (or diffusion) terms, which do not lead to total
energy gain/loss of turbulent energy across the flow field but represent
spatial transports by different effects, namely, advection by the mean flow
and diffusion by turbulence, pressure fluctuation and viscosity. Similarly to
the Reynolds stress transport equation, one can derive the transport equations
of passive-scalar fluxes, and one of the most engineering-relevant features of
turbulence to enhance the spatial transport of heat and mass may be expressed
by the turbulent diffusion terms in these transport equations.

The budget analysis of the Reynolds stress (or turbulent kinetic energy)
transport equation has recently been extended to scale-by-scale budget
analysis, where the transport equation is decomposed into large- and small-
scale parts (e.g., Refs. Kawata and Alfredsson, 2018; Wang, Pan, and Wang,
2020; Doohan, Wills, and Hwang, 2021; Kawata and Tsukahara, 2021; Chan,
Schlatter, and Chin, 2021) or the spectral transport equation Mizuno (2016);
Lee and Moser (2019); Wang  _et al._ (2021) is derived. Similar analysis of
the energy transport at each scale is also possible based on the transport
equations of two-point correlation quantities, such as the second-order
structure function (e.g., Refs. Cimarelli, De Angelis, and Casciola, 2013;
Cimarelli  _et al._ , 2016; Chiarini  _et al._ , 2021). The main feature of
such scale-by-scale budget analysis is that the turbulent diffusion term is
split into the spatial diffusion at each scale and the interscale transport at
each spatial location, which allows us to separately investigate the spatial
and interscale transfer effects by the nonlinear scale interaction of
turbulence.

Such scale-by-scale budget analysis of the turbulent energy transport has been
used to study the complex energy transport in wall turbulence, and the
behaviour of the interscale energy transport term was particularly focused on
in relation to the dynamics of coherent structures in the inner and outer
layers. As these inner and outer structures have clearly different spanwise
length scales Smits, McKeon, and Marusic (2011), many of the earlier studies
on the scale-by-scale budget analysis were based on spanwise length scales
(e.g., Refs. Cho, Hwang, and Choi, 2018; Kawata and Alfredsson, 2018; Chan,
Schlatter, and Chin, 2021; Doohan, Wills, and Hwang, 2021), and it has been
shown that the interscale energy transport terms exhibit both forward (from
larger to smaller scales) and backward (from smaller to larger scales) energy
transfers in the near-wall region Mizuno (2016); Cho, Hwang, and Choi (2018);
Kawata and Alfredsson (2018); Hamba (2018); Chiarini  _et al._ (2021); Doohan,
Wills, and Hwang (2021), unlike in homogeneous isotropic turbulence, where the
energy transfer is basically from larger to smaller scales.

The scale-by-scale analysis of the passive-scalar fluxes can also be performed
by the almost same manner as the analysis of the Reynolds stress transport as
their transport equations are similar. Such scale-by-scale analysis of
passive-scalar transfer allows us to investigate the interscale and spatial
transport of the passive scalar caused by nonlinear interactions between
different scales in the velocity fields. However, despite numerous studies
reported on the wall turbulence with a passive scalar field (e.g., Refs.
Kawamura  _et al._ , 1998; Kawamura, Abe, and Matsuo, 1999; Hane  _et al._ ,
2006; Tsukahara, Iwamoto, and Kawamura, 2007; Antonia and Abe, 2009; Abe  _et
al._ , 2007; Alcántara-Ávila, Hoyas, and Pérez-Quiles, 2018; Alcántara-Ávila,
Gandía-Barberá, and Hoyas, 2019; Alcántara-Ávila, Hoyas, and Jezabel Pérez-
Quiles, 2021; Alcántara-Ávila and Hoyas, 2021) in the last few decades, the
scale-by-scale transport of the passive scalar in wall turbulence has not been
explored yet.

In the present study, we investigate the transport equation budgets of the
temperature-related spectra based on the direct numerical simulation (DNS)
data of wall turbulence with a passive-scalar heat transfer, with a particular
focus on the roles of the interscale and spatial transport at each scale.
Turbulent plane Couette flow is chosen as the test case, because very-large-
scale structures filling up the entire channel appear at relatively low
Reynolds numbers Komminaho, Lundbladh, and Johansson (1996); Kitoh,
Nakabayashi, and Nishimura (2005); Tsukahara, Kawamura, and Shingai (2006);
Kitoh and Umeki (2008); Avsarkisov  _et al._ (2015); Orlandi, Bernardini, and
Pirozzoli (2015); Lee and Moser (2018), and therefore, scale separation
between the inner and outer structures is relatively clearer than in other
canonical wall-bounded flow configurations at the same Reynolds number. The
constant-temperature-difference condition is adopted as the thermal boundary
condition on the walls so that the mean velocity and temperature profiles are
similar to each other. Then, fluctuating velocity and temperature fields are
compared by focusing on the similarity/difference in the interscale and
spatial energy transport caused by scale interactions. In our previous paper
Kawata and Tsukahara (2022), the discussion was mainly based on streamwise
Fourier-mode analysis, and the budget analysis based on spanwise scale was not
focused on. This paper investigates the spectral transport of the temperature
fluctuation and the turbulent heat fluxes based on the spanwise Fourier mode.
In particular, our interest lies in the transport of the wall-normal turbulent
heat flux in comparison with that of the Reynolds shear stress, where inverse
interscale transfer is observed throughout the channel.

##  II  DNS Dataset and Spectral Budget Analysis

###  II.1 DNS Dataset of a Turbulent Plane Couette Flow with Temperature
Transport

In the present study, the budget analysis of the spectral transport equations
is performed with a DNS dataset of a turbulent plane Couette flow with
passive-scalar temperature transport obtained in our previous work Kawata and
Tsukahara (2022). Here, we briefly summarise the computational conditions. The
geometrical configuration of the plane Couette flow is such that a shear flow
is driven by the stationary bottom wall and the top wall translating with a
constant speed $U_{\mathrm{w}}$, which are separated by a distance $h$. The
$x$-, $y$-, and $z$-axes are taken in the streamwise, wall-normal, and
spanwise directions, and the origin of the coordinates is placed on the
stationary bottom wall; the bottom and top walls are located at $y=0$ and
$y=h$, respectively. The temperatures of the walls are uniform and constant in
time, and the temperature difference is $\Delta
T=T_{\mathrm{t}}-T_{\mathrm{b}}$ ($T_{\mathrm{t}}$ and $T_{\mathrm{b}}$ are
the temperature of the top and the bottom walls, respectively).

Table 1: Computational conditions: streamwise and spanwise length of computational domain, $L_{x}^{\ast}$ and $L_{z}^{\ast}$; number of grid points in each direction, $N_{x}$, $N_{y}$, and $N_{z}$; spatial resolutions, $\Delta x$, $\Delta y$, and $\Delta z$; friction Reynolds number $Re_{\tau}$. $\Delta y_{\mathrm{max}}$ and $\Delta y_{\mathrm{min}}$ are the maximum and minimum value of the wall-normal spatial resolution, respectively, and the superscript + for the spatial resolutions stands for the values scaled by viscous length $\nu^{\ast}/u_{\tau}^{\ast}$. The outer Reynolds number and the Prandtl number are $Re_{\mathrm{w}}=8600$ and $Pr=0.71$, respectively.  $(L_{x}^{\ast},L_{z}^{\ast})$ | $(N_{x},N_{y},N_{z})$ | $(\Delta x^{+},\Delta z^{+})$ | $(\Delta y^{+}_{\mathrm{max}},\Delta y^{+}_{\mathrm{min}})$ | $Re_{\tau}$
---|---|---|---|---
$(24h,12.8h)$ | $(512,96,512)$ | $(11.8,6.31)$ | $(6.16,0.268)$ | $126.1$

The governing equations of fluid flow are the continuity and Navier-Stokes
equations for incompressible flow that are non-dimensionalised by $h$ and
$U_{\mathbf{w}}$, and the convection-diffusion equation of passive-scalar
temperature that was by $\Delta T$, $h$, and $U_{\mathbf{w}}$ was solved for
the fluctuating temperature field. A non-slip boundary condition was applied
to the fluid velocity on the wall, and as described above the temperature on
the walls are uniform and constant in time with the temperature difference
$\Delta T$. It should be noted here that both the velocity and temperature
fields are driven by a uniform and constant velocity or temperature difference
between the top and bottom walls, and therefore the boundary conditions on the
walls for the velocity and temperature fields are similar when scaled by the
velocity difference $U_{\mathrm{w}}$ and the temperature difference $\Delta
T$, respectively. For the streamwise and spanwise directions, periodic
boundary conditions were applied. The Reynolds number $Re=U_{\mathrm{w}}h/\nu$
and Prandtl number $Pr=\alpha/\nu$ are $Re_{\mathbf{w}}=8600$ and $Pr=0.71$,
where $\nu$ and $\alpha$ are the kinematic viscosity and the thermal
diffusivity of the fluid, respectively, and other details of the DNS, such as
the domain size and spatial resolution, are listed in Table 1.

Figure 1: Profiles of mean streamwise velocity $U$ and the mean temperature
$\varTheta$ in (top) outer and (bottom) inner scaling. In the bottom panel,
the grey chained lines indicate $U^{+}=y^{+}$ and $U^{+}=2.44\ln y^{+}+5.1$
for the mean velocity profile and $\varTheta^{+}=Pry^{+}$ for the mean
temperature profile. The black dashed line in the panel (b) presents reference
data Kawata and Tsukahara (2021) obtained with an extremely large
computational domain $(L_{x},L_{z})=(96.0h,12.8h)$.

We denote the velocity components in the $x$-, $y$-, and $z$-directions as
$U+u$, $v$, $w$, respectively. Here $U$ is the mean streamwise velocity and
the lower-case letter represents the fluctuation around the mean value of each
velocity component (note here that the mean wall-normal and spanwise
velocities are zero). The temperature is defined as the temperature difference
between the fluid and the bottom wall and denoted as $\varTheta+\theta$: here
$\varTheta$ and $\theta$ are the mean temperature and the temperature
fluctuation, respectively, and $\varTheta=0$ and $\Delta T$ at the bottom and
top walls, respectively. In the following, $\left\langle\right\rangle$
represents averaged quantities obtained by averaging in $x$\- and
$z$-directions and in time. They are also averaged between the lower and upper
half of the channel because of the symmetric flow configuration.

Figures 1 and 2 show the profiles of the mean velocity and temperature and
those of the second-order statistics, such as velocity and temperature
variances and their cross correlations. Here, $T_{\tau}$ used in Fig. 1(b) is
the friction temperature defined as $T_{\tau}=Q_{\mathrm{w}}/\rho
c_{p}u_{\tau}$, where $Q_{\mathrm{w}}$ and $c_{p}$ are the mean heat flux on
the wall and the specific heat at constant pressure, respectively, and
$u_{\tau}$ is the friction velocity defined as
$u_{\tau}=\sqrt{\tau_{\mathrm{w}}/\rho}$ ($\tau_{\mathrm{w}}$ and $\rho$ are
the wall shear stress and the fluid density). As shown in Fig. 1(a), the
profile of the mean temperature $\varTheta$ is anti-symmetric similarly to the
mean velocity profile due to the similar boundary conditions for the velocity
and temperature fields, and it is also seen in Fig. 1(b) that the mean
temperature gradient $\mathrm{d}\varTheta/\mathrm{d}y$ in the vicinity of the
wall is markedly smaller than the mean streamwise velocity gradient
$\mathrm{d}U/\mathrm{d}y$. This difference can be attributed to the fact that
$Pr=0.71<1$, which makes the effect of molecular diffusion more significant in
the temperature field than in the velocity field.

Figure 2: Profiles of (solid lines) the Reynolds stresses $\left\langle
u_{i}u_{j}\right\rangle$ and (dashed lines) the temperature-related turbulent
statistics $\left\langle\theta^{2}\right\rangle$ and $\left\langle
u_{i}\theta\right\rangle$. The values are scaled based on the inner units,
$u_{\tau}$ and/or $T_{\tau}$, and only $0\ll y\ll 0.5$ is shown. The black
circles present reference data for the Reynolds stresses Kawata and Tsukahara
(2021).

Figure 2 presents the profiles of the Reynolds stresses $\left\langle
u^{2}\right\rangle$, $\left\langle v^{2}\right\rangle$, $\left\langle
w^{2}\right\rangle$, and $\left\langle uv\right\rangle$, and the temperature-
related statistics: the temperature fluctuation
$\left\langle\theta^{2}\right\rangle$ and the velocity-temperature
correlations $\left\langle u\theta\right\rangle$ and $\left\langle
v\theta\right\rangle$. As shown here, the temperature fluctuation
$\left\langle\theta^{2}\right\rangle$ and the streamwise-velocity-temperature
correlation $\left\langle u\theta\right\rangle$ have similar profiles to the
streamwise velocity fluctuation $\left\langle u^{2}\right\rangle$, which
indicates a certain similarity between the fluctuations of the streamwise
velocity $u$ and of the temperature $\theta$. This is attributable to the fact
that the profiles of the mean streamwise velocity $U$ and mean temperature
$\varTheta$ are similar. It is also shown here that the profiles of the
Reynolds shear stress $\left\langle uv\right\rangle$ and turbulent heat flux
$\left\langle v\theta\right\rangle$ are almost on top on each other, which
indicates strong similarity between the turbulent momentum and the heat
transfers.

In the DNS of turbulent plane Couette flow, one needs to use a considerably
large computational domain to exclude the dependency of the obtained DNS
results on the computational domain size due to the very-large-scale structure
Komminaho, Lundbladh, and Johansson (1996). The effect of the domain size was
carefully addressed in our previous work Kawata and Tsukahara (2021), and it
was confirmed that the domain size used for obtaining the present DNS dataset,
$(L_{x},L_{z})=(24.0h,12.8h)$, was large enough to exclude the domain-size
effect, as demonstrated in Figs. 1 and 2 by the good agreement between the
present computational results and those obtained with a much larger
computational domain Kawata and Tsukahara (2021).

###  II.2 Spectral Analysis of the Reynolds-Stress Transport and Turbulent
Heat Transfers

In the present study, the spectral transports of the temperature fluctuation
$\left\langle\theta^{2}\right\rangle$ and the turbulent heat fluxes
$\left\langle u_{i}\theta\right\rangle$ are analysed based on spanwise Fourier
mode analysis. The transport equations of the temperature-related spectra are
derived similarly to those for the Reynolds stress spectra. Here, the
derivation of the transport equations of the Reynolds stress spectra is
briefly reviewed, and the spectral transport equations for the temperature-
related statistics are introduced by similar means.

####  II.2.1 Transport of Reynolds stresses

The transport of the Reynolds stresses $\left\langle u_{i}u_{j}\right\rangle$
is expressed as

$\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle
u_{i}u_{j}\right\rangle=P_{ij}-\varepsilon_{ij}+\Phi_{ij}+D^{\nu}_{ij}+D^{t}_{ij},$
(1a) where the terms on the right-hand side are the production ($P_{ij}$),
viscous dissipation ($\varepsilon_{ij}$), pressure-gradient work
($\Phi_{ij}$), viscous diffusion ($D^{\nu}_{ij}$), and turbulent diffusion
($D^{t}_{ij}$) terms, which are defined as $\displaystyle P_{ij}$
$\displaystyle=-\left\langle u_{i}u_{k}\right\rangle\frac{\partial
U_{j}}{\partial x_{k}}-\left\langle u_{j}u_{k}\right\rangle\frac{\partial
U_{i}}{\partial x_{k}},\quad\varepsilon_{ij}=2\nu\left\langle\frac{\partial
u_{i}}{\partial x_{k}}\frac{\partial u_{j}}{\partial x_{k}}\right\rangle,$
(1b) $\displaystyle\Phi_{ij}$ $\displaystyle=-\left\langle\frac{\partial
p}{\partial x_{i}}u_{j}\right\rangle-\left\langle\frac{\partial p}{\partial
x_{j}}u_{i}\right\rangle,\quad D^{\nu}_{ij}=\nu\frac{\partial^{2}\left\langle
u_{i}u_{j}\right\rangle}{\partial x_{k}^{2}},\quad
D^{t}_{ij}=-\frac{\partial\left\langle u_{i}u_{j}u_{k}\right\rangle}{\partial
x_{k}}.$ (1c)

The pressure term $\Phi_{ij}$ may be decomposed into the pressure-strain
correlation $\Pi_{ij}$ and the pressure diffusion $D^{p}_{ij}$:

$\displaystyle\Phi_{ij}=\Pi_{ij}+D^{p}_{ij},\quad\mathrm{where}\quad\Pi_{ij}$
$\displaystyle=\left\langle p\left(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\right\rangle,$ (2a)
$\displaystyle D^{p}_{ij}$ $\displaystyle=-\frac{\partial}{\partial
x_{k}}\left(\left\langle u_{i}p\right\rangle\delta_{jk}+\left\langle
u_{j}p\right\rangle\delta_{ik}\right).$ (2b)

$\delta_{ij}$ is the Kronecker’s delta.

Then, we consider the decomposition of fluctuating velocities and temperature
into their large- and small-scale parts

$\displaystyle u_{i}=u_{i}^{L}+u_{i}^{S},\quad\theta=\theta^{L}+\theta^{S},$
(3a) at a cutoff spanwise wavenumber $k_{z,c}$ based on the spanwise Fourier
modes. With such a decomposition the cross correlation between the large- and
small-scale parts is zero for any combination of quantities:
$\displaystyle\left\langle u_{i}^{L}u_{j}^{S}\right\rangle=\left\langle
u_{j}^{L}u_{i}^{S}\right\rangle=0,\quad\left\langle\theta^{L}\theta^{S}\right\rangle=\left\langle
u_{i}^{L}\theta^{S}\right\rangle=\left\langle
u_{i}^{S}\theta^{L}\right\rangle=0.$ (3b)

Hence, the Reynolds stresses $\left\langle u_{i}u_{j}\right\rangle$ are
decomposed simply into their large- and small-scale parts as

$\displaystyle\left\langle u_{i}u_{j}\right\rangle=\left\langle
u_{i}^{L}u_{j}^{L}\right\rangle+\left\langle u_{i}^{S}u_{j}^{S}\right\rangle,$
(4)

and by a similar manner to derive the “full” Reynolds stress equation (1) one
obtains the transport equations of the large- and small-scale parts of the
Reynolds stresses (the details of the derivation are found in Refs. Kawata and
Alfredsson, 2019; Kawata and Tsukahara, 2021) as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle
u_{i}^{L}u_{j}^{L}\right\rangle=P^{L}_{ij}-\varepsilon^{L}_{ij}+\Phi^{L}_{ij}+D^{\nu,L}_{ij}+D^{t,L}_{ij}-Tr^{z}_{ij},$
(5a) $\displaystyle\left(\frac{\partial}{\partial
t}+U_{k}\frac{\partial}{\partial x_{k}}\right)\left\langle
u_{i}^{S}u_{j}^{S}\right\rangle=P^{S}_{ij}-\varepsilon^{S}_{ij}+\Phi^{S}_{ij}+D^{\nu,S}_{ij}+D^{t,S}_{ij}+Tr^{z}_{ij}.$
(5b)

The terms on the right-hand sides of Eqs. (5a, b) are the large- and small-
scale parts of their counterpart in Eq. (1), except for $Tr_{ij}$. In
particular, the first four terms on the right-hand side of each equation
(i.e., the production, viscous dissipation, pressure-gradinet work, and
viscous diffusion) are simply decomposed into their large- and small-scale
parts similarly to the Reynolds stress decomposition in Eq. (4): these terms
in Eq. (5a) are defined as those in Eq. (1b-e) with $u_{i}$ and $u_{j}$
replaced by $u_{i}^{L}$ and $u_{j}^{L}$ (the corresponding terms in Eq. (5b)
are similarly defined with $u_{i}^{S}$ and $u_{j}^{S}$).

On the other hand, the turbulent diffusion is decomposed as

$\displaystyle D^{t,L}=-\frac{\partial\left\langle
u_{i}u_{j}u_{k}\right\rangle^{L}}{\partial x_{k}},\quad
D^{t,S}=-\frac{\partial\left\langle u_{i}u_{j}u_{k}\right\rangle^{S}}{\partial
x_{k}},$ (6)

where $\left\langle u_{i}u_{j}u_{k}\right\rangle^{L}$ and $\left\langle
u_{i}u_{j}u_{k}\right\rangle^{S}$ are, respectively, the large- and small-
scale part of the triple velocity correlation $\left\langle
u_{i}u_{j}u_{k}\right\rangle$ defined as

$\displaystyle\left\langle u_{i}u_{j}u_{k}\right\rangle^{L}=\left\langle
u_{i}^{L}u_{j}^{L}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{L}u_{j}^{L}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{S}u_{j}^{L}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{L}u_{j}^{S}u_{k}^{S}\right\rangle,$ (7a) $\displaystyle\left\langle
u_{i}u_{j}u_{k}\right\rangle^{S}=\left\langle
u_{i}^{S}u_{j}^{S}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{S}u_{j}^{S}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{L}u_{j}^{S}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{S}u_{j}^{L}u_{k}^{L}\right\rangle,$ (7b)

and the last term in Eqs. (5a, b), $Tr^{z}_{ij}$, is defiend as

$\displaystyle Tr^{z}_{ij}=-\left\langle u_{i}^{S}u_{k}^{S}\frac{\partial
u_{j}^{L}}{\partial x_{k}}\right\rangle-\left\langle
u_{j}^{S}u_{k}^{S}\frac{\partial u_{i}^{L}}{\partial
x_{k}}\right\rangle+\left\langle u_{i}^{L}u_{k}^{L}\frac{\partial
u_{j}^{S}}{\partial x_{k}}\right\rangle+\left\langle
u_{j}^{L}u_{k}^{L}\frac{\partial u_{i}^{S}}{\partial x_{k}}\right\rangle.$ (8)

It is observed from Eqs.(5a, b) that this term appears in the transport
equations of both $\left\langle u_{i}^{L}u_{j}^{L}\right\rangle$ and
$\left\langle u_{i}^{S}u_{j}^{S}\right\rangle$ with different signs, which
means that $Tr^{z}_{ij}$ represents the energy flux between the large- and
small-scale velocity fields across the cutoff wavenumber $k_{z,c}$.

It is worth pointing out here that only the turbulent diffusion terms
($D^{t,L}_{ij}$ and $D^{t,S}_{ij}$) and the interscale energy flux term
($Tr^{z}_{ij}$) include both the large- and small-scale parts of the
fluctuating velocities, whereas the other terms in the transport equations
consist of either the large- or small-scale part only. This is because the
turbulent transport and the interscale energy flux terms are the third-order
moments of the fluctuating velocities (and the velocity gradients), while the
other terms are second-order moments. These terms including both the large-
and small-scale parts of the fluctuating velocity can be interpreted as the
energy transport effects by interactions between different scales: the
turbulent transport ($D^{t,L}_{ij}$ and $D^{t,S}_{ij}$) indicates the energy
transport in physical space caused by scale interactions, while the interscale
energy flux ($Tr^{z}_{ij}$) is the energy transport between different scales.

As the large-scale part of the Reynolds stresses $\left\langle
u_{i}^{L}u_{j}^{L}\right\rangle$ and the spanwise one-dimensional spectra of
the Reynolds stresses are related as

$\displaystyle E^{z}_{ij}=\frac{\partial\left\langle
u_{i}^{L}u^{L}_{j}\right\rangle}{\partial
k_{z,c}}\left(=-\frac{\partial\left\langle
u_{i}^{S}u_{j}^{S}\right\rangle}{\partial k_{z,c}}\right),$ (9)

the transport equations of the Reynolds-stress spectra can be derived by
differentiating both sides of Eq. (5a) with respect to $k_{z,c}$ as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{ij}=pr^{z}_{ij}-\xi^{z}_{ij}+\phi^{z}_{ij}+d^{\nu,z}_{ij}+d^{t,z}_{ij}+tr^{z}_{ij},$
(10)

where the terms on the right-hand side are the $k_{z,c}$-derivatives of the
counter parts in Eq. (5a). The first five terms represent the spectra of the
corresponding terms of Eq.( 1a): the spectra of the production
($pr^{z}_{ij}$), viscous dissipation ($\xi^{z}_{ij}$), pressure term
($\phi^{z}_{ij}$), viscous diffusion ($d^{\nu,z}_{ij}$) and turbulent
diffusion ($d^{t,z}_{ij}$) terms. The interscale transport term $tr^{z}_{ij}$
represents the energy gain/loss by the interscale flux $Tr_{ij}$ at each
wavenumebr $k_{z}$. According to Eq. (2), the pressure-gradient-work spectrum
$\phi_{ij}$ can also be decomposed into the spectra of the pressure-strain
energy redistribution term $\Pi_{ij}$ and the pressure diffusion term
$D^{p}_{ij}$ as

$\displaystyle\phi^{z}_{ij}=\pi^{z}_{ij}+d^{p,z}_{ij}.$ (11)

The turbulent spatial transport $d^{t,z}_{ij}$ and the interscale transport
$tr^{z}_{ij}$, which are of particular interest in the present study, are
defined as

$\displaystyle d^{z,t}_{ij}=-\frac{\partial E^{z}_{ijk}}{\partial
x_{k}},\quad\quad tr^{z}_{ij}=-\frac{\partial Tr^{z}_{ij}}{\partial k_{z,c}}.$
(12a) Here, $E^{z}_{ijk}$ is the spectra of triple velocity correlations
$\left\langle u_{i}u_{j}u_{k}\right\rangle$ defined as $\displaystyle
E^{z}_{ijk}=\frac{\partial\left\langle
u_{i}u_{j}u_{k}\right\rangle^{L}}{\partial
k_{z,c}}=-\frac{\partial\left\langle
u_{i}u_{j}u_{k}\right\rangle^{S}}{\partial k_{z,c}}.$ (12b)

As the physical meaning of $\left\langle u_{i}u_{j}u_{k}\right\rangle$ is the
spatial transport flux of the Reynolds stress $\left\langle
u_{i}u_{j}\right\rangle$ in the $x_{k}$-direction caused by the velocity
fluctuation $u_{k}$, their spectra $E^{z}_{ijk}$ represent the spatial flux of
the Reynolds stresses $\left\langle u_{i}u_{j}\right\rangle$ in the
$x_{k}$-direction at each spanwise wavenumber.

####  II.2.2 Transport of temperature fluctuation and velocity-temperature
correlations

Now, we derive the spectral transport equations of the temperature-related
turbulence statistics. The transport equation of the temperature fluctuation
$\left\langle\theta^{2}\right\rangle$ is written similarly to the Reynolds
stress equation (1) as

$\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle\theta^{2}\right\rangle=P_{\theta\theta}-\varepsilon_{\theta\theta}+D^{\nu}_{\theta\theta}+D^{t}_{\theta\theta},$
(13a) with the production ($P_{\theta\theta}$), viscous dissipation
($\varepsilon_{\theta\theta}$), viscous diffusion ($D^{\nu}_{\theta\theta}$),
and turbulent transport ($D^{t}_{\theta\theta}$) terms defined as
$\displaystyle P_{\theta\theta}=-2\left\langle\theta
u_{k}\right\rangle\frac{\partial\varTheta}{\partial
x_{k}},\quad\varepsilon_{\theta\theta}=2\alpha\left\langle\frac{\partial\theta}{\partial
x_{k}}\frac{\partial\theta}{\partial x_{k}}\right\rangle,\quad
D^{\nu}_{ij}=\alpha\frac{\partial^{2}\left\langle\theta^{2}\right\rangle}{\partial
x_{k}^{2}},\quad
D^{t}_{\theta\theta}=-\frac{\partial\left\langle\theta^{2}u_{k}\right\rangle}{\partial
x_{k}}.$ (13b)

The transport equation of the turbulent heat fluxes $\left\langle
u_{i}\theta\right\rangle$ is also similarly obtained as

$\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle
u_{i}\theta\right\rangle=P_{i\theta}-\varepsilon_{i\theta}+\Phi_{i\theta}+D^{\nu}_{i\theta}+D^{t}_{i\theta},$
(14a) with the production ($P_{i\theta}$), viscous dissipation
($\varepsilon_{i\theta}$), pressure-gradient-temperature correlations
($\Phi_{i\theta}$), viscous diffusion ($D^{\nu}_{i\theta}$), and turbulent
diffusion ($D^{t}_{i\theta}$) terms defined as $\displaystyle P_{i\theta}$
$\displaystyle=-\left\langle u_{k}\theta\right\rangle\frac{\partial
U_{i}}{\partial x_{k}}-\left\langle
u_{i}u_{k}\right\rangle\frac{\partial\varTheta}{\partial
x_{k}},\quad\varepsilon_{i\theta}=\left(\nu+\alpha\right)\left\langle\frac{\partial
u_{i}}{\partial x_{k}}\frac{\partial\theta}{\partial
x_{k}}\right\rangle,\quad\Phi_{i\theta}=-\left\langle\frac{\partial
p}{\partial x_{i}}\theta\right\rangle,$ (14b) $\displaystyle
D^{\nu}_{i\theta}$ $\displaystyle=\frac{\partial}{\partial
x_{k}}\left(\nu\left\langle\frac{\partial u_{i}}{\partial
x_{k}}\theta\right\rangle+\alpha\left\langle\frac{\partial\theta}{\partial
x_{k}}u_{i}\right\rangle\right),\quad\quad
D^{t}_{i\theta}=-\frac{\partial\left\langle u_{i}\theta
u_{k}\right\rangle}{\partial x_{k}}.$ (14c)

Based on the decomposition given by Eq. (3), the temperature variance and the
turbulent heat fluxes are also split into their large- and small-scale parts,
similarly to the Reynolds stress decomposition (4), as

$\displaystyle\left\langle\theta^{2}\right\rangle=\left\langle\theta^{L}\theta^{L}\right\rangle+\left\langle\theta^{S}\theta^{S}\right\rangle,\quad\left\langle
u_{i}\theta\right\rangle=\left\langle
u_{i}^{L}\theta^{L}\right\rangle+\left\langle
u_{i}^{S}\theta^{S}\right\rangle.$ (15)

Therefore, the transport equations of their large- and small-scale parts can
be derived similarly to Eqs. (5). Then, these large- and small-scale parts of
the temperature-related statistics are related to their spectra, similarly to
Eq. (9), as

$\displaystyle
E^{z}_{\theta\theta}=\frac{\partial\left\langle\theta^{L}\theta^{L}\right\rangle}{\partial
k_{z,c}},\quad E^{z}_{i\theta}=\frac{\partial\left\langle
u_{i}^{L}\theta^{L}\right\rangle}{\partial k_{z,c}},$ (16)

and the transport equations of these temperature-related spectra can also be
obtained by differentiating the equations of
$\left\langle\theta^{L}\theta^{L}\right\rangle$ and $\left\langle
u_{i}^{L}\theta^{L}\right\rangle$. With such procedures one obtains the
temperature-variance spectrum equation as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{\theta\theta}=pr^{z}_{\theta\theta}-\xi^{z}_{\theta\theta}+d^{\nu,z}_{\theta\theta}+d^{t,z}_{\theta\theta}+tr^{z}_{\theta\theta},$
(17)

with the terms on the right-hand side representing the spectra of the
counterparts in Eq. (13a). The interscale and the turbulent spatial transport
terms are defined as

$\displaystyle tr^{z}_{\theta\theta}=-\frac{\partial
Tr^{z}_{\theta\theta}}{\partial k_{z,c}},\quad\quad
d^{t,z}_{\theta\theta}=-\frac{\partial E^{z}_{\theta\theta k}}{\partial
x_{k}},$ (18a) where $Tr^{z}_{\theta\theta}$ and $E^{z}_{\theta\theta k}$ are
the interscale and spectral spatial fluxes of the temperature fluctuation
defined as $\displaystyle Tr^{z}_{\theta\theta}$
$\displaystyle=-2\left\langle\theta^{S}u_{k}^{S}\frac{\partial\theta^{L}}{\partial
x_{k}}\right\rangle-2\left\langle\theta^{L}u_{k}^{L}\frac{\partial\theta^{S}}{\partial
x_{k}}\right\rangle,$ (18b) $\displaystyle E^{z}_{\theta\theta k}$
$\displaystyle=\frac{\partial\left\langle\theta^{2}u_{k}\right\rangle^{L}}{\partial
k_{z,c}}=-\frac{\partial\left\langle\theta^{2}u_{k}\right\rangle^{S}}{\partial
k_{z,c}},$ (18c) with $\left\langle\theta^{2}u_{k}\right\rangle^{L}$ and
$\left\langle\theta^{2}u_{k}\right\rangle^{S}$ being the large- and small-
scale part of the spatial flux of temperature fluctuation
$\left\langle\theta^{2}u_{k}\right\rangle$:
$\displaystyle\left\langle\theta^{2}u_{k}\right\rangle^{L}$
$\displaystyle=\left\langle\theta^{L}\theta^{L}u_{k}^{L}\right\rangle+\left\langle\theta^{L}\theta^{L}u_{k}^{S}\right\rangle+2\left\langle\theta^{L}\theta^{S}u_{k}^{S}\right\rangle,$
(18d) $\displaystyle\left\langle\theta^{2}u_{k}\right\rangle^{S}$
$\displaystyle=\left\langle\theta^{S}\theta^{S}u_{k}^{S}\right\rangle+\left\langle\theta^{S}\theta^{S}u_{k}^{L}\right\rangle+2\left\langle\theta^{L}\theta^{S}u_{k}^{L}\right\rangle.$
(18e)

Similarly, the transport equation of the turbulent heat flux spectrum
$E^{z}_{i\theta}$ is obtained as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{i\theta}=pr^{z}_{i\theta}-\xi^{z}_{i\theta}+\phi^{z}_{i\theta}+d^{\nu,z}_{i\theta}+d^{t,z}_{i\theta}+tr^{z}_{i\theta},$
(19)

and the terms on the right-hand side are, again, the spectra of their
counterparts in Eq. (14a). The interscale and the turbulent spatial transport
terms are defined in the same manner as in other spectral transport equations:

$\displaystyle tr^{z}_{i\theta}=-\frac{\partial Tr^{z}_{i\theta}}{\partial
k_{z,c}},\quad\quad d^{t,z}_{i\theta}=-\frac{\partial E^{z}_{i\theta
k}}{\partial x_{k}},$ (20a) where the interscale flux $Tr^{z}_{i\theta}$ and
spectral spatial flux $E^{z}_{i\theta k}$ are, respectively, defined as
$\displaystyle Tr^{z}_{i\theta}$
$\displaystyle=-\left\langle\theta^{S}u_{k}^{S}\frac{\partial
u_{i}^{L}}{\partial x_{k}}\right\rangle-\left\langle
u_{i}^{S}u_{k}^{S}\frac{\partial\theta^{L}}{\partial
x_{k}}\right\rangle+\left\langle\theta^{L}u_{k}^{L}\frac{\partial
u_{i}^{S}}{\partial x_{k}}\right\rangle+\left\langle
u_{i}^{L}u_{k}^{L}\frac{\partial\theta^{S}}{\partial x_{k}}\right\rangle,$
(20b) $\displaystyle E^{z}_{i\theta k}$
$\displaystyle=\frac{\partial\left\langle u_{i}\theta
u_{k}\right\rangle^{L}}{\partial k_{z,c}}=-\frac{\partial\left\langle
u_{i}\theta u_{k}\right\rangle^{S}}{\partial k_{z,c}}.$ (20c) Here,
$\left\langle u_{i}\theta u_{k}\right\rangle^{L}$ and $\left\langle
u_{i}\theta u_{k}\right\rangle^{S}$ are the large- and small-scale parts of
$\left\langle u_{i}\theta u_{k}\right\rangle$ defined as
$\displaystyle\left\langle u_{i}\theta u_{k}\right\rangle^{L}=\left\langle
u_{i}^{L}\theta^{L}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{L}\theta^{L}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{L}\theta^{S}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{S}\theta^{L}u_{k}^{S}\right\rangle,$ (20d) $\displaystyle\left\langle
u_{i}\theta u_{k}\right\rangle^{S}=\left\langle
u_{i}^{S}\theta^{S}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{S}\theta^{S}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{S}\theta^{L}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{L}\theta^{S}u_{k}^{L}\right\rangle.$ (20e)

##  III Results

###  III.1 Spectra of Temperature-Related Statistics and Their Interscale and
Spatial Transports

Figure 3 presents the distributions of the spanwise one-dimensional spectra of
the streamwise turbulent energy $E^{z}_{uu}$, temperature fluctuation
$E^{z}_{\theta\theta}$, and temperature-velocity correlation $E^{z}_{u\theta}$
in the premultiplied form. The distribution of the streamwise turbulent energy
spectrum $E^{z}_{uu}$ in the panel (a) shows two significant energy
concentrations, one of which is the energy peak located in a relatively small
wavelength range near the wall around $(y^{+},\lambda_{z}^{+})\approx(12,100)$
and the other is the broad energy band at a large wavelength around
$\lambda_{z}/h\approx 2.5$ covering most of the channel. These energy peaks
clearly correspond to the small-scale coherent structures near the wall and
the very-large-scale structure of plane Couette turbulence, respectively. As
shown in this figure, the energy peaks corresponding to these coherent
structures at different scales are observed to be clearly separated from each
other despite the relatively small Reynolds number $Re_{\tau}=126$, which is a
unique feature of plane Couette turbulence Kitoh, Nakabayashi, and Nishimura
(2005); Tsukahara, Kawamura, and Shingai (2006); Kitoh and Umeki (2008).

Figure 3: Space-wavelength ($y$-$\lambda_{z}$) diagrams of spanwise one-
dimensional premultiplied spectra of (a) streamwise turbulent energy
$k_{z}E^{z}_{uu}$, (b) temperature fluctuation $k_{z}E^{z}_{\theta\theta}$,
and (c) velocity-temperature correlation $k_{z}E^{z}_{u\theta}$. The values
are scaled by $u_{\tau}^{2}$, $T_{\tau}^{2}$, and $u_{\tau}T_{\tau}$,
respectively.

The temperature-fluctuation spectrum $E^{z}_{\theta\theta}$ presents a similar
distribution to the streamwise turbulent energy $E^{z}_{uu}$ as shown in Fig.
3(b), in which the energy peaks corresponding to the near-wall and very-large-
scale structures are indicated at almost the same position in the
$y$-$\lambda_{z}$ diagram. The location of the near-wall peak is, however, at
a slightly larger wavelength and further from the wall than the corresponding
peak in the $E^{z}_{uu}$ distribution,
$(y^{+},\lambda_{z}^{+})\approx(16,134)$, which can be attributed to the more
significant effect of molecular diffusion in the temperature field as $Pr<1$.
The distribution of the temperature-velocity cospectrum $E^{z}_{u\theta}$ also
shows a similar distribution to $E^{z}_{uu}$, indicating similar behaviours of
the fluctuating streamwise velocity and temperature.

Figure 4: Spanwise one-dimensional spectra of (a) the Reynolds shear stress
$E^{z}_{-uv}$ and (b) turbulent heat transfer $E^{z}_{-v\theta}$, presented in
the same manner as in Fig. 3.

Figure 4 compares the spanwise one-dimensional cospectra of the Reynolds shear
stress $E^{z}_{-uv}$ and the velocity-temperature correlation
$E^{z}_{-v\theta}$. As the Reynolds shear stress $-\left\langle
uv\right\rangle$ and the temperature-velocity correlation $-\left\langle
v\theta\right\rangle$ represent the wall-normal transport of momentum and heat
by turbulent fluid motions, respectively, their cospectra represent the
turbulent momentum and heat transfers at each length scale. As shown in Fig.
4(a), the distribution of the Reynolds shear stress cospectrum $E^{z}_{-uv}$
presents both the inner peak at small scales near the wall and the broad
energy peak at large scales at the channel centre, similarly to the
distribution of the streamwise turbulent energy spectrum $E^{z}_{uu}$. As
shown in Fig. 4(b), the distribution of the turbulent heat transfer spectrum
$E^{z}_{-v\theta}$ is qualitatively similar to that of the momentum transfer
spectrum $E^{z}_{-uv}$, presenting both the inner and outer energy peaks
corresponding to the coherent structures in the near-wall and channel-central
regions of the channel, which indicates a certain scale-by-scale similarity
between the momentum and heat transfers by turbulence.

###  III.2 Interscale Fluxes of the Reynolds Stresses and Temperature-Related
Statistics

Figure 5: Space-wavelength ($y$-$\lambda_{z}$) diagrams of spanwise interscale
fluxes of (a) the turbulent kinetic energy $Tr^{z}_{uu}$, (b) temperature
fluctuation $Tr^{z}_{\theta\theta}$, and (c) temperature-velocity correlation
$Tr^{z}_{u\theta}$. The values are scaled by $u_{\tau}^{4}/\nu$,
$u_{\tau}^{2}$, $T_{\tau}^{2}/\nu$, and $u_{\tau}^{3}T_{\tau}/\nu$,
respectively.

Next, the interscale fluxes of the temperature-related spectra are
investigated. Figure 5 presents the distributions of the spanwise interscale
flux of the streamwise turbulent energy $Tr^{z}_{uu}$, temperature fluctuation
$Tr^{z}_{\theta\theta}$, and temperature-velocity correlation
$Tr^{z}_{u\theta}$. Here, as can be seen from Eq. (5), the positive values
indicate forward energy fluxes (i.e., from larger to smaller scales) in the
spanwise length-scale direction, whereas the negative values represent the
inverse (from smaller to larger scales) energy fluxes. As shown in Fig. 5(a),
the turbulent energy flux $Tr^{z}_{uu}$ indicates mainly forward interscale
energy transfers from larger to smaller $\lambda_{z}$ in the relatively small
$\lambda_{z}$ range throughout the channel, but it also presents backward
energy fluxes from smaller to larger $\lambda_{z}$ in a relatively large
$\lambda_{z}$ range in the near-wall region. It is interesting to note that
that such inverse interscale energy transfer is not observed in the streamwise
length-scale direction Lee and Moser (2019); Kawata and Tsukahara (2021), and
the corresponding physical phenomenon has still not been elucidated.

As shown in Fig. 5(b), the spanwise interscale flux of the temperature
fluctuation $Tr^{z}_{\theta\theta}$ presents a qualitatively similar
distribution to the streamwise turbulent energy flux $Tr^{z}_{uu}$, presenting
a region of backward interscale energy transfers at relatively large
$\lambda_{z}$ in the near-wall region and forward interscale energy fluxes at
smaller $\lambda_{z}$ throughout the channel. One can, however, also observe
in the distribution of $Tr^{z}_{\theta\theta}$ that the region of backward
energy flux somewhat shrinks and instead the magnitude of the forward energy
flux around the channel centre is relatively stronger than $Tr^{z}_{uu}$.

Figure 5(c) shows the distribution of the spanwise interscale flux of the
velocity-temperature correlation $Tr^{z}_{u\theta}$. One can observe that
$Tr^{z}_{u\theta}$ also has a qualitatively similar distribution to those of
the turbulent energy flux $Tr^{z}_{uu}$ and the temperature fluctuation flux
$Tr^{z}_{\theta\theta}$, presenting both backward and forward energy flux.
Furthermore, as compared to the distributions of $Tr^{z}_{uu}$ and
$Tr^{z}_{\theta\theta}$, the region of backward energy flux is relatively
large and the peak magnitude of the forward energy flux at small $\lambda_{z}$
near the wall is relatively small.

The spanwise interscale fluxes of the Reynolds shear stress $Tr^{z}_{-uv}$ and
the turbulent heat flux $Tr^{z}_{-v\theta}$ are also presented in Fig. 6. As
shown in the panel (a), the Reynolds shear stress flux $Tr^{z}_{-uv}$
indicates, interestingly, the backward transfers throughout the channel, as
first experimentally observed by Kawata & Alfredsson Kawata and Alfredsson
(2018). This is in contrast to the turbulent energy transfer $Tr^{z}_{uu}$,
which mainly indicates forward transfers throughout the channel except the
near-wall region. Such inverse interscale transport of the Reynolds shear
stress has also been reported in turbulent channel Chiarini  _et al._ (2021)
and boundary-layer Chan, Schlatter, and Chin (2021) flows.

Figure 6: Spanwise interscale fluxes of (a) Reynolds shear stress
$Tr^{z}_{-uv}$ and (b) turbulent heat flux $Tr^{z}_{-v\theta}$, presented in
the same manner as in Fig. 5.

The interscale transfer of the turbulent heat flux $Tr^{z}_{-v\theta}$ gives a
similar distribution to that of $Tr^{z}_{-uv}$, as presented in Fig. 6(b). The
difference between the distributions of these spanwise interscale fluxes is
that $Tr^{z}_{-v\theta}$ spans to slightly larger spanwise wavelengths than
$Tr^{z}_{-uv}$. While it is still unclear what physical phenomenon is
represented by such an inverse interscale transfer of $-\left\langle
uv\right\rangle$ and $-\left\langle v\theta\right\rangle$, the similar
distributions of $Tr^{z}_{-uv}$ and $Tr^{z}_{-v\theta}$ indicate close analogy
between the transport of the Reynolds shear stress and turbulent heat flux.

###  III.3 Spectra of Spatial Turbulent Fluxes of the Reynolds Stresses and
Temperature-Related Statistics

We focus on the spatial turbulent transport of the Reynolds stresses and
temperature-related statistics. Figure 7 presents the profiles of the triple
velocity correlations $\left\langle u^{2}v\right\rangle$ and $-\left\langle
uv^{2}\right\rangle$ and those of the triple velocity-temperature correlations
$\left\langle\theta^{2}v\right\rangle$ and $-\left\langle
v^{2}\theta\right\rangle$. As for the physical interpretations of these third-
order statistics, $\left\langle u^{2}v\right\rangle$ and
$\left\langle\theta^{2}v\right\rangle$ represent respectively the wall-normal
spatial fluxes of the velocity fluctuation $\left\langle u^{2}\right\rangle$
and the temperature fluctuation $\left\langle\theta^{2}\right\rangle$, which
are caused by turbulent fluid motions. Similarly, $-\left\langle
uv^{2}\right\rangle$ and $-\left\langle v^{2}\theta\right\rangle$ indicate the
turbulent spatial fluxes of the Reynolds shear stress $-\left\langle
uv\right\rangle$ and the turbulent heat transfer $-\left\langle
v\theta\right\rangle$, respectively. As discussed in Section II.2, these
third-order statistics represent the spatial transport effect caused by
interactions between different scales. As shown in Fig. 7, the profiles of the
triple correlations are all qualitatively similar, indicating transport
further towards the wall in the near-wall region and transport towards the
channel centre in the far-wall region.

Figure 7: Profiles of the velocity triple correlations $\left\langle
u^{2}v\right\rangle$ and $-\left\langle uv^{2}\right\rangle$ and the
temperature-velocity triple correlations
$\left\langle\theta^{2}v\right\rangle$ and $-\left\langle
v^{2}\theta\right\rangle$. The values are scaled by $u_{\tau}^{3}$,
$u_{\tau}T_{\tau}^{2}$, or $u_{\tau}^{2}T_{\tau}$.  Figure 8: Spanwise one-
dimensional premultiplied spectra of turbulent spatial fluxes of (a)
streamwise turbulent energy $k_{z}E^{z}_{uuv}$, (b) temperature fluctuation
$k_{z}E^{z}_{\theta\theta v}$, and (c) temperature-velocity correlation
$k_{z}E^{z}_{u\theta v}$, presented in the same manner as in Fig. 3. The
values are scaled by $u_{\tau}^{3}$, $u_{\tau}T_{\tau}^{2}$, and
$u_{\tau}^{2}T_{\tau}$, respectively.

Figure 8 presents the distributions of the spanwise one-dimensional spectra of
the turbulent energy transports $E^{z}_{uuv}$, $E^{z}_{\theta\theta v}$, and
$E^{z}_{u\theta v}$ in the premultiplied form. These spectra represent the
spectral contents of the triple correlations $\left\langle
u^{2}v\right\rangle$, $\left\langle\theta^{2}v\right\rangle$, and
$\left\langle u\theta v\right\rangle$ presented in Fig. 7. As shown in Fig. 8,
these spectra of spatial energy fluxes show similar distributions, indicating
that the negative (i.e., towards the wall) spatial transport in the near-wall
region mainly occurs at the largest wavelength $\lambda_{z}/h\approx 2.5$,
while the positive (towards the channel centre) transport occurs at two
smaller scales: the middle scale roughly at $\lambda_{z}/h\approx 0.8$ and
relatively small wavelengths around $\lambda_{z}/h\approx 0.2$
($\lambda_{z}^{+}\approx 50$). As the wavelength of the negative
$k_{z}E^{z}_{uuv}$ peak at the largest scale $\lambda_{z}/h=2.5$ coincides
with the typical spanwise spacing of $u$-streaks of the very-large-scale
structure, the large-scale negative peak presumably represents the turbulence
transports in which the fluctuating velocities and temperature are carried
towards the wall vicinity by the secondary fluid motions of the very-large-
scale structures. As for the positive $k_{z}E^{z}_{uuv}$ peak at a relatively
small scale $\lambda_{z}^{+}\approx 50$, this wavelength fairly matches the
peak location of the premultiplied wall-normal turbulent energy spectrum
$k_{z}E^{z}_{vv}$ in the near-wall region Tsukahara, Kawamura, and Shingai
(2006), which is likely related to the coherent fluid motions in the near-wall
region. Therefore, it can be inferred that this positive $k_{z}E^{z}_{uuv}$
peak indicates the turbulent energy transport driven by the wall-normal fluid
motions by the near-wall coherent structures. On the other hand, the
wavelength of the positive $k_{z}E^{z}_{uuv}$ peak on the larger-scale side,
$\lambda_{z}/h\approx 0.8$ ($\lambda_{z}^{+}\approx 100$), is on the order of
the full channel height $h$, but any corresponding characteristic length scale
of coherent fluid motion is not found. As this scale lies in the middle
wavelength range between the length scales of the very-large-scale and near-
wall structures, this positive $k_{z}E^{z}_{uuv}$ peak may represent the
energy transport caused by interactions between these inner and outer coherent
structures.

Figure 9: Spanwise one-dimensional premultiplied spectra of turbulent spatial
fluxes of (a) Reynolds shear stress $E^{z}_{-uvv}$ and (b) the turbulent heat
flux $E^{z}_{-v\theta v}$, presented in the same manner as in Fig. 3. The
values are scaled by $u_{\tau}^{3}$ and $u_{\tau}^{2}T_{\tau}$, respectively.

Figure 9 presents the distributions of the spatial flux spectra of the
Reynolds shear stress $E^{z}_{-uvv}$ and the turbulent heat transfer
$E^{z}_{-v\theta v}$. Similarly to other spatial flux spectra presented in
Fig. 8, $E^{z}_{-uvv}$ and $E^{z}_{-v\theta v}$ are the spectral contents of
the triple correlations $-\left\langle uv^{2}\right\rangle$ and $-\left\langle
v^{2}\theta\right\rangle$ presented in Fig. 7, indicating the wall-normal
fluxes of the Reynolds shear stress $-\left\langle uv\right\rangle$ and the
turbulent heat flux $-\left\langle v\theta\right\rangle$ at each wavelength
$\lambda_{z}$, respectively. As shown in the figure, these spatial flux
spectra of the cross correlations present qualitatively similar distributions
to those of the spatial turbulent energy fluxes, such as $E^{z}_{uuv}$ and
$E^{z}_{\theta\theta v}$ presented in Fig. 8, indicating the peak of negative
transport at the largest wavelength $\lambda_{z}/h\approx 2.5$ near the wall
and the positive transport peaks at two smaller wavelengths in the relatively
far-wall region. This tendency is in contrast to the interscale fluxes in that
the interscale fluxes of the cross correlations $Tr^{z}_{-uv}$ and
$Tr^{z}_{-v\theta}$ show clearly different behaviours from those of turbulent
energies such as $Tr^{z}_{uu}$ and $Tr^{z}_{\theta\theta}$, as observed in
Figs. 5 and 6.

As described so far, the spectral distributions of the fluctuating velocity
and temperature fields have been investigated with respect to their interscale
and spatial fluxes, and it has been observed that the temperature-related
statistics essentially show similar distributions to the corresponding
turbulence statistics, indicating a close similarity between the fluctuating
velocity and temperature fields. In the next section, the spectral budgets of
the transport equations of the temperature-related statistics are investigated
further in detail.

###  III.4 Budget Analysis of Spectral Transport Equations

Now, we investigate the transport budget of the temperature-related spectra
$E^{z}_{\theta\theta}$, $E^{z}_{u\theta}$, and $E^{z}_{-v\theta}$ in detail.
The transport budgets of the Reynolds stress spectra are also given in Apendix
A, and compared to these temperature-related spectra transport when necessary
for discussion. Figure 10 presents the distributions of terms in the transport
equation of the temperature fluctuation spectrum $E^{z}_{\theta\theta}$:

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{\theta\theta}=pr^{z}_{\theta\theta}-\xi^{z}_{\theta\theta}+d^{\nu,z}_{\theta\theta}+d^{t,z}_{\theta\theta}+tr^{z}_{\theta\theta}.$
(21)

As shown in the top-left panel, the temperature fluctuation is produced by
$pr^{z}_{\theta\theta}$ mainly at a relatively small wavelength of
$\lambda_{z}^{+}\approx 100$ in the near-wall region, which roughly
corresponds to the $y$-position and the spanwise length scale of the coherent
structure in the near-wall region. This tendency is quite similar to the
distribution of the streamwise turbulent energy production spectrum
$pr^{z}_{uu}$ given in Fig. 15, see Appendix A. The similarity between these
productions can be understood by noting that $pr^{z}_{\theta\theta}$ and
$pr^{z}_{uu}$ depend on the cospectra $E^{z}_{-v\theta}$ and $E^{z}_{-uv}$,
respectively, as

$\displaystyle
pr^{z}_{\theta\theta}=E^{z}_{-v\theta}\frac{\mathrm{d}\varTheta}{\mathrm{d}y},\quad
pr^{z}_{uu}=E^{z}_{-uv}\frac{\mathrm{d}U}{\mathrm{d}y},$ (22)

and the distributions of $E^{z}_{-v\theta}$ and $E^{z}_{-uv}$ and the profiles
of $U$ and $\varTheta$ are similar to each other, respectively, as presented
in Figs. 1 and 4.

Figure 10: Space-wavelength ($y$-$\lambda_{z}$) diagrams of terms in the
transport equation of the temperature fluctuation spectrum
$E^{z}_{\theta\theta}$: (top left) production $pr^{z}_{\theta\theta}$; (top
right) viscous terms $\xi^{z}_{\theta\theta}+d^{\nu,z}_{\theta\theta}$;
(bottom left) interscale transport $tr^{z}_{\theta\theta}$; (bottom right)
turbulent diffusion $d^{t,z}_{\theta\theta}$. The terms are presented in the
premultiplied form, and the values are scaled by
$u_{\tau}^{2}T_{\tau}^{2}/\nu$.

The distribution of the interscale transport $tr^{z}_{\theta\theta}$ presents
the consequential energy gain/loss by the interscale energy flux
$Tr^{z}_{\theta\theta}$ shown in Fig. 5 as $tr^{z}_{\theta\theta}=-\partial
Tr^{z}_{\theta\theta}/\partial k_{z}$, and it is shown in Fig. 10 that the
energy is removed from the region around the peak of $pr^{z}_{\theta\theta}$
and transported towards both larger and smaller $\lambda_{z}$ ranges in the
near-wall region by the interscale energy transport $tr^{z}_{\theta\theta}$,
while in the channel-core region the energy is mainly transported in the
forward direction from around $\lambda_{z}/h\approx 2.5$ to a smaller
$\lambda_{z}$ range.

The distribution of the spectral turbulent diffusion $d^{t,z}_{\theta\theta}$
represents the energy gain/loss by the turbulent spatial flux
$E^{z}_{\theta\theta v}$ presented in Fig. 8, as
$d^{t,z}_{\theta\theta}=-\partial E^{z}_{\theta\theta v}/\partial y$. Its
contribution is relatively small as compared to the other terms, but it can be
seen that the energy is supplied to the wall vicinity at a large wavelength
$\lambda_{z}/h\approx 2.5$, and the energy transport towards the channel
central region is also observed at two relatively small wavelength ranges,
corresponding to the distribution of the turbulent spatial flux
$E^{z}_{\theta\theta v}$. The energy produced by $pr^{z}_{\theta\theta}$ and
transported by $tr^{z}_{\theta\theta}$ and $d^{t,z}_{\theta\theta}$ is
eventually dissipated by the viscous dissipation $\xi^{z}_{\theta\theta}$.
Such distributions of the terms in the $E^{z}_{\theta\theta}$ transport
equation described above are quite similar to those of the corresponding terms
in the $E^{z}_{uu}$ transport equation given in Fig. 15.

The clear difference between the transport equations of the temperature
fluctuation spectrum $E^{z}_{\theta\theta}$ and the streamwise turbulent
energy $E^{z}_{uu}$ is that no pressure-related term exists in the
$E^{z}_{\theta\theta}$ transport equation, whereas in the $E^{z}_{uu}$
transport the pressure-strain cospectrum $\pi^{z}_{uu}$ plays an important
role to redistribute energy from $E^{z}_{uu}$ to other components of turbulent
energy spectra. Despite such a distinct difference, the distributions of the
terms in the $E^{z}_{\theta\theta}$ equation are quite similar to those of the
corresponding terms of the $E^{z}_{uu}$ transport equation (compare Figs. 10
and 15). In the following, the budget balances of the $E^{z}_{\theta\theta}$
and $E^{z}_{uu}$ equations are compared in more detail.

Figure 11: Detailed budget balance of the $E^{z}_{\theta\theta}$ and
$E^{z}_{uu}$ transport equation at (a) a near-wall location $y^{+}=16$ and (b)
the quarter height of the channel $y/h=0.25$: (blue) production; (red) viscous
dissipation$+$viscous diffusion; (yellow) interscale transport; (purple)
turbulent diffusion; (light green) pressure-strain cospectrum $\pi^{z}_{uu}$.
The solid and dashed lines represent terms in the $E^{z}_{\theta\theta}$ and
$E^{z}_{uu}$ equations, respectively. The terms are given in the premultiplied
form, and the values are scaled by the peak value of each production term.

Figure 11 presents detailed balance between the terms of the
$E^{z}_{\theta\theta}$ and $E^{z}_{uu}$ equations at a near-wall location
$y^{+}=16$ and the quarter height of the channel $y/h=0.25$ ($y^{+}\approx
60$). The values are scaled by the maximum value of each production term. As
shown in the panel (a), the profiles of the production, interscale transport,
and turbulent diffusion of the $E^{z}_{\theta\theta}$ and $E^{z}_{uu}$
transport equations show a good collapse in the near-wall region. The
production terms indicate a significant energy gain at $\lambda_{z}^{+}\approx
100$, as already pointed out above, and the interscale transport terms are
shown to transfer the energy from this energy-producing wavelength to both
larger and smaller $\lambda_{z}$ ranges, where the turbulent diffusion terms
transfer some energy to other $y$-positions. As previously mentioned, there is
no term in the $E^{z}_{\theta\theta}$ transport budget corresponding to the
pressure-strain term $\pi^{z}_{uu}$ in the $E^{z}_{uu}$ transport, and the
consequential difference between the budget balances of the
$E^{z}_{\theta\theta}$ and $E^{z}_{uu}$ transport equation is mainly
manifested by the viscous terms. As there is no energy absorption by the
pressure-strain cospectrum in the $E^{z}_{\theta\theta}$ budget, the viscous
terms $\xi^{z}_{\theta\theta}+d^{\nu,z}_{\theta\theta}$ show a clearly larger
contribution than $\xi^{z}_{uu}+d^{\nu,z}_{uu}$ to compensate for the absence
of the pressure term and, thus, the transport equation is balanced.

At the quarter height of the channel $y/h=0.25$, as shown in Fig. 11(b), the
profiles of the production terms collapse similarly to those in the near-wall
region, but the $\lambda_{z}$ range where energy is dissipated by viscosity is
significantly smaller than the energy-producing $\lambda_{z}$ range. Hence,
the interscale transport terms bridge these energy-producing and -dissipating
$\lambda_{z}$ ranges by forward energy transfer and do not exhibit reversed
energy transfer, unlike in the near-wall region. It is also observed that the
production terms present not only the largest peak of energy gain at
$\lambda_{z}^{+}\approx 200$ but also a secondary peak at
$\lambda_{z}/h\approx 2.5$, which corresponds to the very-large-scale
structures. The turbulent diffusion terms present energy gains at two
wavelength ranges, which represents energy supply from the near-wall region
(see the $d^{t,z}_{\theta\theta}$ and $d^{t,z}_{uu}$ distributions in Figs. 10
and 15). The contribution of the pressure-strain correlation $\pi^{z}_{uu}$ in
the $E^{z}_{uu}$ transport equation is relatively large compared to that in
the near-wall location shown in the panel (a), and the interscale transport
$tr^{z}_{\theta\theta}$ as well as the viscous terms
$\xi^{z}_{\theta\theta}+d^{\nu,z}_{\theta\theta}$ show significantly larger
contributions than the counterparts in the $E^{z}_{uu}$ equation to compensate
for the absence of the energy absorption by $\pi^{z}_{uu}$. Thus, both in the
near- and far-wall regions of the channel, the energy productions in the
$E^{z}_{\theta\theta}$ and $E^{z}_{uu}$ transports are quite similar to each
other, but more energy is dissipated by the viscous terms in the
$E^{z}_{\theta\theta}$ transport, compensating for the absence of the energy
loss by the pressure-related term.

Figure 12: Distribution of terms in the transport equation of the velocity-
temperature cospectrum $E^{z}_{u\theta}$ presented in the same manner as in
Fig. 10: (top left) production $k_{z}pr^{z}_{u\theta}$; (top centre) viscous
terms $k_{z}(\xi^{z}_{u\theta}+d^{\nu,z}_{u\theta})$; (top right) pressure
term $k_{z}\phi^{z}_{u\theta}$; (bottom right) interscale transport
$tr^{z}_{u\theta}$; (bottom centre) turbulent diffusion $d^{t,z}_{u\theta}$.
The values are scaled by $u_{\tau}^{3}T_{\tau}/\nu$.

The transport equation of the velocity-temperature cospectrum
$E^{z}_{u\theta}$ is written as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{u\theta}=pr^{z}_{u\theta}-\xi^{z}_{u\theta}+d^{\nu,z}_{u\theta}+\phi^{z}_{u\theta}+d^{t,z}_{u\theta}+tr^{z}_{u\theta},$
(23)

and the distributions of the terms in the right-hand side are presented in
Fig. 12. It is clearly seen here that the $E^{z}_{u\theta}$ transport equation
has a pressure term $\phi^{z}_{u\theta}$ and it essentially functions as an
energy sink throughout the channel, similarly to the pressure-strain energy
redistribution term $\pi^{z}_{uu}$ in the $E^{z}_{uu}$ transport equation. It
is also noteworthy that the production of the velocity-temperature cospectrum
$pr^{z}_{u\theta}$ depends on both the Reynolds shear stress spectra
$E^{z}_{-uv}$ and the turbulent heat flux spectra $E^{z}_{-v\theta}$ as

$\displaystyle
pr^{z}_{u\theta}=E^{z}_{-v\theta}\frac{\mathrm{d}U}{\mathrm{d}y}+E^{z}_{-uv}\frac{\mathrm{d}\varTheta}{\mathrm{d}y},$
(24)

unlike the productions of other spectra, such as $pr^{z}_{uu}$ and
$pr^{z}_{\theta\theta}$, which only depend on either $E^{z}_{-uv}$ or
$E^{z}_{-v\theta}$. The distribution of $pr^{z}_{u\theta}$ is, however,
similar to those of $pr^{z}_{uu}$ and $pr^{z}_{\theta\theta}$, as the
distributions of $E^{z}_{-uv}$ and $E^{z}_{-v\theta}$ are similar to each
other and so are the profiles of $U$ and $\varTheta$. Each term of the
$E^{z}_{u\theta}$ transport equation presented in Fig. 12 shows a similar
distribution to the corresponding term in the $E^{z}_{uu}$ transport equation
shown in Fig. 15. Since the $E^{z}_{u\theta}$ transport equation has a
pressure-related term, unlike the $E^{z}_{\theta\theta}$ equation, the budget
balance is even closer to the transport of the streamwise turbulent energy
$E^{z}_{uu}$ than that of the $E^{z}_{\theta\theta}$ transport.

Now, we shed light on the transport of the turbulent heat flux spectrum
$E^{z}_{-v\theta}$. The transport equation of $E^{z}_{-v\theta}$ is written as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{-v\theta}=pr^{z}_{-v\theta}-\xi^{z}_{-v\theta}+d^{\nu,z}_{-v\theta}+\phi^{z}_{-v\theta}+d^{t,z}_{-v\theta}+tr^{z}_{-v\theta},$
(25)

and the distributions of the terms on the right-hand side are presented in
Fig. 13. As shown here, $E^{z}_{-v\theta}$ is produced by the production term
$pr^{z}_{-v\theta}$ mainly in the near-wall region at wavelengths around
$\lambda_{z}^{+}\approx 100$, similarly to the other energy spectra
$pr^{z}_{\theta\theta}$ and $pr^{z}_{u\theta}$. The energy gain by the
production $pr^{z}_{-v\theta}$ is mainly balanced by the pressure term
$\phi^{z}_{-v\theta}$, and the viscous terms do not play an important role.

Figure 13: Distribution of terms in the transport equation of the turbulent
heat flux spectrum $E^{z}_{-v\theta}$ presented in the same manner as in Fig.
12: (top left) production $k_{z}pr^{z}_{-v\theta}$; (top centre) viscous terms
$k_{z}(\xi^{z}_{-v\theta}+d^{\nu,z}_{-v\theta})$; (top right) pressure term
$k_{z}\phi^{z}_{-v\theta}$; (bottom left) interscale transport
$tr^{z}_{-v\theta}$; (bottom centre) turbulent diffusion $d^{t,z}_{-v\theta}$.
The values are scaled by $u_{\tau}^{3}T_{\tau}/\nu$.

The turbulent transport terms $tr^{z}_{-v\theta}$ and $d^{t,z}_{-v\theta}$ are
shown to partly transport $E^{z}_{-v\theta}$ in scale and space, respectively.
The interscale transport $tr^{z}_{-v\theta}$ presents the energy gain/loss by
the inverse interscale flux $Tr^{z}_{-v\theta}$ investigated in Fig. 6, and
the distribution shows that the turbulent heat flux $-\left\langle
v\theta\right\rangle$ is mainly transported from wavelengths around
$\lambda_{z}/h\approx 0.25$ ($\lambda_{z}^{+}\approx 65$) to a larger
$\lambda_{z}$ range around $\lambda_{z}/h\approx 0.75$
($\lambda_{z}^{+}\approx 190$) in the near-wall region. The distribution of
$d^{t,z}_{-v\theta}$ indicates weak spatial transports similarly to
$d^{t,z}_{\theta\theta}$ and $d^{t,z}_{u\theta}$, where $-\left\langle
v\theta\right\rangle$ is supplied to the wall vicinity from the channel-core
region at the largest wavelengths $\lambda_{z}/h\approx 2.5$ and at a smaller
$\lambda_{z}$ range $-\left\langle v\theta\right\rangle$ is transported from
the near- to far-wall region at two different wavelengths.

The tendencies of the terms in the $E^{z}_{-v\theta}$ transport equation
described above are quite similar to those of the corresponding terms in the
transport equation of the Reynolds shear stress spectrum $E^{z}_{-uv}$, as can
be seen by comparing Fig. 13 with Fig. 18 in Appendix A. It is particularly
noteworthy here that the productions of $E^{z}_{-v\theta}$ and $E^{z}_{-uv}$
are both dependent on the wall-normal turbulent energy spectrum $E^{z}_{vv}$
as

$\displaystyle
pr^{z}_{-v\theta}=E^{z}_{vv}\frac{\mathrm{d}\varTheta}{\mathrm{d}y},\quad
pr^{z}_{-uv}=E^{z}_{vv}\frac{\mathrm{d}U}{\mathrm{d}y},$ (26)

which indicates that the Reynolds shear stress $-\left\langle uv\right\rangle$
and the turbulent heat flux $-\left\langle v\theta\right\rangle$ are produced
at the same scale.

Figure 14: Detailed budget balance of the $E^{z}_{-v\theta}$ and $E^{z}_{-uv}$
transport equation at (a) a near-wall location $y^{+}=16$ and (b) the quarter
height of the channel $y/h=0.25$ presented in the same manner as in Fig. 11:
(blue) production; (red) viscous dissipation$+$viscous diffusion; (yellow)
interscale transport; (purple) turbulent diffusion; (light green) pressure
term.

Figure 14 presents a detailed comparison of the transport budgets of
$E^{z}_{-v\theta}$ and $E^{z}_{-uv}$ at a near-wall location $y^{+}=16$ and
the quarter height of the channel in the same manner as in Fig. 11. Note here
that in both panels the profiles of the productions $pr^{z}_{-v\theta}$ and
$pr^{z}_{-uv}$ are exactly on top of each other since their profiles are
similar, as shown by Eq. (26). As shown here, at both wall-normal locations,
the spectral budget balances of the $E^{z}_{-v\theta}$ and $E^{z}_{-uv}$
transport are quite similar to each other. It is also shown at both wall-
normal locations that the peaks of the pressure terms are located at a larger
$\lambda_{z}$ than those of the production terms in both $E^{z}_{-v\theta}$
and $E^{z}_{-uv}$ transport, which indicates that the turbulent heat flux
$-\left\langle v\theta\right\rangle$ and the Reynolds shear stress
$-\left\langle uv\right\rangle$ are dissipated by the effect of pressure at
larger spanwise wavelengths than those at which they are produced by the mean
temperature or velocity gradients. Such tendencies are in contrast to the
transport of the energy spectra such as $E^{z}_{uu}$ and
$E^{z}_{\theta\theta}$, where the energy is dissipated basically at smaller
scales than produced. This is attributable to the contribution by the
interscale transport terms $tr^{z}_{-v\theta}$ and $tr^{z}_{-uv}$ to transport
the energy from smaller to larger scales.

It is interesting to note that the agreements between the turbulent diffusion
terms $d^{t,z}_{-v\theta}$ and $d^{t,z}_{-uv}$ is particularly good while a
slight difference can be found between the interscale transport terms
$tr^{z}_{-v\theta}$ and $tr^{z}_{-uv}$, although both the turbulent diffusion
and interscale transport represent the transport effect caused by nonlinear
interactions between different scales. Such good agreement between the
turbulent diffusion terms are also observed between the $E^{z}_{\theta\theta}$
and $E^{z}_{uu}$ transports, as already pointed out with Fig. 11.

##  IV Discussion and Concluding Remarks

In the previous section, the spectral budgets of the temperature-related
spectra, such as $E^{z}_{\theta\theta}$, $E^{z}_{u\theta}$, and
$E^{z}_{-v\theta}$, were compared with those of the corresponding Reynolds
stress spectra, $E^{z}_{uu}$ and $E^{z}_{-uv}$, and close analogy between
those turbulence transports was indicated. Of particular interest is the close
similarity between the spectral transports of the Reynolds shear stress
$-\left\langle uv\right\rangle$ and the turbulent heat flux $-\left\langle
v\theta\right\rangle$, where inverse interscale transport from a smaller to
larger $\lambda_{z}$ is observed.

The turbulent diffusion and interscale transport are two different aspects of
nonlinear multi-scale interactions of turbulence, and the physical phenomena
these spatial and interscale transport terms represent is of great interest.
As shown in Fig. 14(a), the turbulent diffusions $d^{t,z}_{-v\theta}$ and
$d^{t,z}_{-uv}$ are shown to remove energy from the near-wall location in two
$\lambda_{z}$ ranges around $\lambda_{z}^{+}\approx 50$ and $180$, and the
removed $-\left\langle v\theta\right\rangle$ and $-\left\langle
uv\right\rangle$ are partly transported towards the central region of the
channel, as indicated by the profiles of $d^{t,z}_{-v\theta}$ and
$d^{t,z}_{-uv}$ at $y/h=0.25$ given in Fig. 14(b). In particular, the energy
gains by $d^{t,z}_{-v\theta}$ and $d^{t,z}_{-uv}$ at $y/h=0.25$ are shown to
balance well with the energy loss by the interscale energy transport
$tr^{z}_{-v\theta}$ and $tr^{z}_{-uv}$, which suggests that the energy
($-\left\langle v\theta\right\rangle$ and $-\left\langle uv\right\rangle$)
spatially transferred from the near-wall region to this wall-normal location
at relatively small $\lambda_{z}$ is further transferred to a larger
$\lambda_{z}$ range. Kawata & Alfredsson Kawata and Alfredsson (2018)
experimentally observed similar Reynolds shear stress transport from smaller
$\lambda_{z}$ near the wall to larger $\lambda_{z}$ in the channel-core region
in turbulent plane Couette flow and conjectured that it may represent the
influences of the near-wall smaller-scale coherent structures on the very-
large-scale structures, which is similar to the concept of a co-supporting
cycle of the inner and outer structures proposed by Toh & Itano Toh and Itano
(2005).

It should be mentioned here that the inverse interscale transport of the
Reynolds shear stress and turbulent heat flux described so far are observed
based on one-dimensional spanwise-Fourier-mode analysis, and therefore, the
interscale energy transfer in the streamwise length scales is not
investigated. In fact, such reversed interscale transfers are not observed by
the analysis based on the streamwise Fourier modes. In a numerical simulation
by Kawata & Tsukahara Kawata and Tsukahara (2021), spectral budget analysis
was performed based on both one-dimensional streamwise and spanwise Fourier
modes, and it was shown that the interscale transfers of the turbulent energy
and Reynolds shear stress in the streamwise wavenumber direction are basically
forward transfers (i.e., from larger to smaller scales). Lee & Moser Lee and
Moser (2019) performed two-dimensional Fourier-mode analysis on the turbulent
energy transport and showed that the energy transport between wavenumbers with
the same magnitude but different directions, which may not represent energy
transfer between really different length scales, can be observed as inverse
interscale energy transfers through one-dimensional Fourier mode analysis.
These observations suggest that the inverse energy transfers in the spanwise
length scales may not simply be interpreted as interaction from smaller- to
larger-scale structures.

Some recent studies, on the other hand, investigated the relation between the
inner-outer interaction in wall turbulence and the inverse interscale energy
transfers in the spanwise length scales in detail. Doohan et al. Doohan,
Wills, and Hwang (2021) investigated the interactions between self-sustaining
processes (SSPs) of coherent structures near and away from the wall and argued
that energy transfer from the smaller-scale SSP leads to the formation of the
wall-reaching part of streaks of the larger-scale SSP. Chiarini et al.
Chiarini  _et al._ (2021) also investigated spatial and interscale energy
fluxes by analysing the anisotropic generalised Kolmogorov equation taking
into account both the streamwise and spanwise length scale and observed some
energy paths from smaller scales near the wall to larger scales away from the
wall. Chan et al. Chan, Schlatter, and Chin (2021) also observed the inverse
interscale transport of the Reynolds shear stress in a turbulent boundary
layer, and based on a detailed quadrant analysis they interpreted it as the
net energy transfer from the small-scale ejection (Q2) and sweep (Q4) events
to their large-scale counterparts. Thus, what the inverse interscale energy
transports in the spanwise lengths scales is still a subject of intense
debate. Nevertheless, the transport of the turbulent heat flux spectrum
$E^{z}_{-v\theta}$ is found to be quite similar to that of the Reynolds shear
stress spectrum $E^{z}_{-uv}$, which indicates a close scale-by-scale
similarity between the turbulent momentum and heat transfers.

In the present study, the transport budgets of temperature-related spectra,
such as the temperature-variance and turbulent heat flux spectra, were
investigated in turbulent plane Couette flow with a passive-scalar heat
transfer at the Reynolds number $Re_{\tau}=126$ and the Prandtl number
$Pr=0.71$ based on DNS data. It was found that the transport budgets of the
temperature-related spectra present quite similar tendencies to those of the
corresponding Reynolds stress spectra, including the spectral transport of the
Reynolds shear stress and turbulent heat flux, where inverse interscale
transfers are observed throughout the channel. It is also noteworthy that the
distributions of the spatial flux spectra are all similar, regardless the
spatial fluxes of turbulent energies such as $\left\langle u^{2}\right\rangle$
and $\left\langle\theta^{2}\right\rangle$ or those of cross correlations such
as $-\left\langle uv\right\rangle$ and $-\left\langle v\theta\right\rangle$.
This is in contrast to that the interscale fluxes of $-\left\langle
uv\right\rangle$ and $-\left\langle v\theta\right\rangle$ exhibit notably
different behaviours from those of turbulent energies such as $\left\langle
u^{2}\right\rangle$ and $\left\langle\theta^{2}\right\rangle$. Our next task
is to elucidate what physical phenomena these spatial and interscale
turbulence transport represent. The present investigation is limited to single
values of the Reynolds and Prandtl number, and therefore, the Reynolds- and
Prandtl-number dependency of these interscale and spatial fluxes should also
be investigated in the future studies.

##  Appendix A Spectral Budgets of the Reynolds Stress Transports

Figure 15: Distribution of terms in the transport equation of the streamwise
turbulent energy spectrum $E^{z}_{uu}$, presented in the same manner as in
Fig. 12: (top left) production $k_{z}pr^{z}_{uu}$; (top centre) viscous terms
$k_{z}(\xi^{z}_{uu}+d^{\nu,z}_{uu})$; (top right) pressure-strain energy
redistribution $k_{z}\pi^{z}_{uu}$; (bottom left) interscale transport
$tr^{z}_{uu}$; (bottom centre) turbulent diffusion $d^{t,z}_{uu}$. The values
are scaled by $u_{\tau}^{4}/\nu$. Figure 16: Distribution of terms in the
transport equation of the wall-normal turbulent energy spectrum $E^{z}_{vv}$,
presented in the same manner as in Fig. 12: (top left) pressure-strain energy
redistribution $k_{z}\pi^{z}_{ww}$; (top centre) viscous terms
$k_{z}(-\xi^{z}_{ww}+d^{\nu,z}_{ww})$; (top right) pressure diffusion
$k_{z}d^{p,z}_{vv}$; (bottom left) interscale transport $tr^{z}_{vv}$; (bottom
right) turbulent diffusion $d^{t,z}_{vv}$. The values are scaled by
$u_{\tau}^{4}/\nu$.

Here, the transport budgets of the Reynolds stress spectra are presented for
comparison to those of the temperature-related spectra given in Figs. 10-14.
The transport equations of the streamwise turbulent energy $E^{z}_{uu}$, wall-
normal turbulent energy $E^{z}_{vv}$, and spanwise turbulent energy
$E^{z}_{ww}$ are, respectively, written as

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{uu}$
$\displaystyle=pr^{z}_{uu}-\xi^{z}_{uu}+d^{\nu,z}_{uu}+\pi^{z}_{uu}\hskip
25.6073pt+d^{t,z}_{uu}+tr^{z}_{uu},$ (27)
$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{vv}$ $\displaystyle=\hskip
19.91684pt-\xi^{z}_{vv}+d^{\nu,z}_{vv}+\pi^{z}_{vv}+d^{p,z}_{vv}+d^{t,z}_{vv}+tr^{z}_{vv},$
(28) $\displaystyle\left(\frac{\partial}{\partial
t}+U_{k}\frac{\partial}{\partial x_{k}}\right)E^{z}_{ww}$
$\displaystyle=\hskip
19.91684pt-\xi^{z}_{ww}+d^{\nu,z}_{ww}+\pi^{z}_{ww}\hskip
19.91684pt+d^{t,z}_{ww}+tr^{z}_{ww},$ (29)

and the distributions of each term on the right-hand side of these transport
equations are given in Fig. 15-17. Among the turbulent energy spectra, only
the streamwise component $E^{z}_{uu}$ has energy production from the mean
flow, and the energy sources for the wall-normal and spanwise components
$E^{z}_{vv}$ and $E^{z}_{ww}$ are the energy redistribution by the pressure-
strain cospectra $\pi^{z}_{vv}$ and $\pi^{z}_{ww}$, respectively. Note here
that

$\displaystyle\pi^{z}_{uu}+\pi^{z}_{vv}+\pi^{z}_{ww}=0$ (30)

at any $y$-location and any wavelength $\lambda_{z}$.

Figure 17: Distribution of terms in the transport equation of the spanwise
turbulent energy spectrum $E^{z}_{ww}$, presented in the same manner as in
Fig. 12: (top left) pressure-strain energy redistribution $k_{z}\pi^{z}_{ww}$;
(top right) viscous terms $k_{z}(-\xi^{z}_{ww}+d^{\nu,z}_{ww})$;(bottom left)
interscale transport $tr^{z}_{ww}$; (bottom right) turbulent diffusion
$d^{t,z}_{ww}$. The values are scaled by $u_{\tau}^{4}/\nu$. Figure 18:
Distribution of terms in the transport equation of the Reynolds shear stress
spectrum $E^{z}_{-uv}$, presented in the same manner as in Fig. 12: (top left)
production $k_{z}pr^{z}_{-uv}$; (top centre) viscous terms
$k_{z}(-\xi^{z}_{-uv}+d^{\nu,z}_{-uv})$; (top right) pressure term
$k_{z}\phi^{z}_{-uv}$; (bottom left) interscale transport $tr^{z}_{-uv}$;
(bottom centre) turbulent diffusion $d^{t,z}_{-uv}$. The values are scaled by
$u_{\tau}^{4}/\nu$.

The transport equation of the Reynolds shear stress spectrum is

$\displaystyle\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)E^{z}_{-uv}=pr^{z}_{-uv}-\xi^{z}_{-uv}+d^{\nu,z}_{-uv}+\phi^{z}_{-uv}+d^{t,z}_{-uv}+tr^{z}_{-uv},$
(31)

and the distributions of the budget terms are shown in Fig. 18.

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