# Spectral analysis on dissimilarity between turbulent momentum and heat
transfers 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

Nonlinear interaction between different scales in turbulence results in both
interscale and spatial transport of turbulent energy, and the role of such
scale interactions in turbulent heat transfer mechanism is also of practical
importance from the engineering viewpoint. In the present study, we
investigate a turbulent plane Couette flow with passive-scalar heat transfer
at the Prandtl number 0.71, in order to address the similarity/difference
between the scale interactions in the velocity and temperature fields. The
constant-temperature-difference boundary condition is used so that the mean
velocity and temperature profiles are similar, and then the roles of
interscale and spatial transports are compared for the spectral transport
budgets of the turbulent energies and temperature-related statistics. We show
that turbulent heat transfer occurs at relatively small streamwise length
scales compared to momentum transfer, although molecular diffusion is more
significant in the temperature field as the Prandtl number is less than 1.
Detailed analysis on the transport budgets of the temperature-related spectra
shows that scale interactions in temperature field supply more energy to small
scales than those in velocity field. Such significant temperature cascade
leads to more energetic temperature fluctuation at small scales, which
eventually results in the dissimilarity between the spectra of turbulent heat
and momentum transfers.

††preprint: AIP/123-QED

##  I  Introduction

Nonlinear interaction between different scales is the important aspect of
turbulence, by which turbulent energy is transported across different scales
and spatial diffusions of momentum and heat are enhanced. In wall turbulence
at certainly high Reynolds numbers, the flow involves relatively small-scale
coherent structures in the near-wall region and the large-scale structures
located in the logarithmic region or further out (see, for example, Ref.
Smits, McKeon, and Marusic, 2011 and references therein), and nonlinear
interaction between such inner and outer structures has also been pointed out
in a number of studied in the last decades Mathis, Hutchins, and Marusic
(2009); Bernardini and Pirozzoli (2011); Pirozzoli, Bernardini, and Orlandi
(2011); Dogan  _et al._ (2019).

Motivated by such observations of coherent structures and their interactions,
there has been intensive efforts in recent years to investigate the scale
interactions in wall turbulence by quantifying turbulent energy transfers
between different scales Cimarelli  _et al._ (2016); Mizuno (2016);
Mamatsashvili  _et al._ (2016); Cho, Hwang, and Choi (2018); Kawata and
Alfredsson (2018); Bauer, von Kameke, and Wagner (2019); Hamba (2019); Lee and
Moser (2019); Motoori and Goto (2019, 2020); Chan, Schlatter, and Chin (2021);
Chiarini  _et al._ (2021); Doohan, Wills, and Hwang (2021); Kawata and
Tsukahara (2021). Interestingly, both forward (from larger to smaller scales)
and inverse (from smaller to larger scales) turbulent energy cascades have
been observed in these studies, and the relations of such interscale energy
transfers to the dynamics of coherent structures, such as the self-sustaining
cycles of each inner and outer structureCho, Hwang, and Choi (2018); Hamba
(2019); Wang  _et al._ (2021); Kawata and Tsukahara (2021), the inner-outer
interactionsCho, Hwang, and Choi (2018); Kawata and Alfredsson (2018); Kawata
and Tsukahara (2021); Doohan, Wills, and Hwang (2021); Chiarini  _et al._
(2021); Mäteling and Schröder (2022); Chan  _et al._ (2022); Zhou, Xu, and
Jiménez (2022), merging/splitting of wall-attached hair-pin vorticesChan,
Schlatter, and Chin (2021), etc. have been discussed.

Although only the interscale energy transfer was focused on in the earlier
studies mentioned above, the effect of nonlinear scale interactions to enhance
spatial diffusions is also an important feature of turbulence, particularly in
terms of engineering point of view. One of the most desired controls is the
suppressing momentum transfer with enhancing heat transfer at the same time,
but such dissimilar control of turbulence is usually difficult as the heat
transfer is basically passive to the turbulent fluid motions and therefore
there is a certain analogy between the turbulent heat and momentum transfers.

Some earlier studies, however, demonstrated the possibility of the dissimilar
control of turbulent heat and momentum transfers. Hasegawa & KasagiHasegawa
and Kasagi (2011) and Yamamoto et al.Yamamoto, Hasegawa, and Kasagi (2013)
introduced suboptimal control theory and found that the dissimilar control
works even in flows where the averaged energy and momentum transport equations
have an identical form, and the optimal input mode exhibits a streamwise
traveling-wave-like property. Successful dissimilar controls by traveling-
wave-like input from the wall have been actually reported by some numerical
studies using blowing/suction from the wall(Higashi, Mamori, and Fukagata,
2011; Kaithakkal, Kametani, and Hasegawa, 2020; Mamori  _et al._ , 2021) or
wall deformationUchino, Mamori, and Fukagata (2017). The essential mechanism
of such dissimilar transfer control is the different sensitivities of velocity
and temperature fields to the control input from the wall, which results from
the fact that velocity is a divergence-free vector quantity whereas
temperature is a scalarHasegawa and Kasagi (2011). Deepening our understanding
of such differences in the behaviours of velocity and temperature fields may
allow us to further develop the dissimilar flow control schemes.

Despite numerous studies on wall turbulence with passive-scalar temperature
field reported in the past including aforementioned attempts for the
dissimilar flow control, less attention has been paid to the spectra and
cascades of fluctuating temperature (or passive scalar). Among such few
studies, Antonia & AbeAntonia and Abe (2009) compares the spectra of the
turbulent kinetic energy and the scalar temperature fluctuation in turbulent
channel flows and reported no significant difference between those energy
spectra, but their dissipation spectra display qualitatively different
behaviours. Hane et al.Hane  _et al._ (2006) simulated a turbulent plane
Couette flow with heat transfer and showed that the spectra of the streamwise
turbulent energy and temperature fluctuation indicate their peaks at different
streamwise wavelengths. These studies suggest that the scales dominating the
heat and momentum transfers may differ in spite of the inherent analogy. As
the turbulent heat and momentum transfers are the results of fluid motions in
a wide range of scale, investigating their mechanisms scale-by-scale should
help us, for example, develop new control strategies for heat transfer
enhancement.

In the present study, we investigate the transport equation budgets of the
temperature-related spectra based on the direct numerical simulation data of
wall turbulence with passive-scalar temperature field, with a particular focus
on the roles of the interscale and spatial transports at each scale. Turbulent
plane Couette flow is chosen for the test case, because very-large-scale
structures filling up the entire channel appear even at relatively low
Reynolds numbersBech  _et al._ (1995); 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 the scale
separation between the inner and outer structures is relatively clear as
compared to other canonical wall-bounded flow configurations at similar
Reynolds numbers. The constant-temperature-difference boundary condition is
applied to the temperature field so that the mean velocity and temperature
profiles are similar to each other, and then fluctuating velocity and
temperature fields are compared focusing on similarity/difference in the
interscale and spatial energy transports caused by scale interactions.

##  II  Numerical simulation and data analysis

###  II.1 Direct Numerical Simulation of Plane Couette Flow with Passive-
Scalar Temperature Transport

In the configuration of the plane Couette flow simulated in this study, a flow
between two parallel flat plates separated by distance $h^{\ast}$ (hereafter
the superscript ∗ indicates dimensional quantities) is driven by translation
of the top wall with a constant velocity $U^{\ast}_{\mathrm{w}}$ (the bottom
wall is stationary). The $x^{\ast}$-, $y^{\ast}$\- and $z^{\ast}$-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 $y^{\ast}=0$ and $y^{\ast}=h^{\ast}$, respectively. The
temperatures of the flat plates are uniform and constant in time with the
temperature difference $\Delta
T^{\ast}=T^{\ast}_{\mathrm{t}}-T^{\ast}_{\mathrm{b}}$ ($T^{\ast}_{\mathrm{t}}$
and $T^{\ast}_{\mathrm{b}}$ are the temperature of the top and the bottom
wall, respectively).

The governing equations for the fluid flow simulation are the non-dimensional
continuity and Navier-Stocks equation for incomplessible fluid flow scaled by
$U_{\mathrm{w}}^{\ast}$ and $h^{\ast}$:

$\displaystyle\nabla\cdot\tilde{\bf u}$ $\displaystyle={\bf 0},$ (1a)
$\displaystyle\frac{\partial\tilde{\bf u}}{\partial t}+(\tilde{\bf
u}\cdot\nabla)\tilde{\bf u}$
$\displaystyle=-\nabla\tilde{p}+\frac{1}{Re_{\mathrm{w}}}{\nabla}^{2}\tilde{\bf
u}.$ (1b)

Here the tilde ($\tilde{\;\;}$) indicates an instantaneous quantity including
both the mean and the fluctuating components, and $\tilde{\bf u}$ and
$\tilde{p}$ are the instantaneous velocity vector and pressure of the fluid,
respectively. $Re_{\mathrm{w}}=U_{\mathrm{w}}^{\ast}h^{\ast}/\nu^{\ast}$ is
the Reynolds number ($\nu^{\ast}$ is the kinematic viscosity). The governing
equation of the fluctuating temperature field is

$\displaystyle\frac{\partial\tilde{\theta}}{\partial t}+(\tilde{\bf
u}\cdot\nabla)\tilde{\theta}=\frac{1}{Re_{\mathrm{w}}Pr}{\nabla}^{2}\tilde{\theta},$
(2)

where $Pr$ is the Prandtl number and $\tilde{\theta}$ is the instantaneous
non-dimensional fluid temperature defined as

$\displaystyle\tilde{\theta}=\frac{\tilde{\theta}^{\ast}}{\Delta
T^{\ast}}\quad(\tilde{\theta}^{\ast}=\tilde{T}^{\ast}-T_{b}^{\ast}).$ (3)

Here $\tilde{T}^{\ast}$ is the dimensional instantaneous fluid temperature. In
the present computation, the values of the Reynolds and Prandtl numbers are
$Re_{\mathrm{w}}=8600$ and $Pr=0.71$, respectively.

The present computation is based on our previous study on turbulent plane
Couette flowKawata and Tsukahara (2021). Hence, the numerical procedures to
solve the governing equations for fluid motion (1) are the same, and the
equation for the temperature field (2) is additionally solved with the same
numerically schemes as used for Eqs. (1). The details of the numerical
procedures have already been described elsewhere (for example, Ref. Tsukahara,
Kawamura, and Shingai, 2006), and are therefore only given here briefly. The
governing equations (1) and (2) are solved in physical space with spatial
discretization of the finite difference method, while the pressure Poisson
equation is solved in Fourier space. Central difference schemes with fourth
and second order are used for the wall-parallel ($x$\- and $z$-) and the wall-
normal ($y$-) directions, respectively. Periodic boundary condition is applied
to both the velocity and temperature fields for $x$\- and $z$-directions, and
similar constant-difference conditions are applied to the velocity and
temperature fields on the walls:

$\displaystyle\tilde{\bf u}={\bf 0},\quad\tilde{\theta}=0\quad\text{at}\quad
y=0,$ (4a) $\displaystyle\tilde{\bf
u}=(1,0,0),\quad\tilde{\theta}=1\quad\text{at}\quad y=1.$ (4b)

Due to the existence of the very-large-scale structures in the turbulent plane
Couette flows one needs to use very large computational domains to exclude the
confinement effects on the simulated flow field by the periodic boundary
conditions(Komminaho, Lundbladh, and Johansson, 1996; Tsukahara, Kawamura, and
Shingai, 2006). In our previous study (Kawata and Tsukahara, 2021) we
carefully examined the domain-size effect, and the computational conditions,
such as the domain size and the spatial resolutions in $x$-, $y$\- and
$z$-directions, in the present numerical simulation are the same as in the
Case A1 in the previous study (see Table 1), which was confirmed to give
computational results in a quite good agreement with those obtained with an
extremely large domain ($L_{x}^{\ast}=96h$ and $L_{z}^{\ast}=12.8h$). With the
above-described numerical setup the outer-scale Reynolds number
$Re_{\mathrm{w}}=8600$ results in the friction Reynolds number
$Re_{\tau}=u_{\tau}^{\ast}\delta^{\ast}/\nu^{\ast}=126.0$ (here
$\delta^{\ast}=h^{\ast}/2$ is the channel half height and $u^{\ast}_{\tau}$ is
the friction velocity defined as
$u^{\ast}_{\tau}=\sqrt{\tau^{\ast}_{\mathrm{w}}/\rho^{\ast}}$ with
$\tau^{\ast}_{\mathrm{w}}$ and $\rho^{\ast}$ being the wall shear stress and
the fluid density, respectively), as also given in Table 1.

Table 1: Computational conditions: the streamwise and spanwise domain lengths, the number of grid points, the spatial resolutions, and the friction Reynolds number $Re_{\tau}$. The outer Reynolds number and the Prandtl number are $Re_{\mathrm{w}}=8600$ and $Pr=0.71$, respectively. $(L_{x}^{\ast},L_{z}^{\ast})$ | $(N_{x},N_{y},N_{z})$ | $(\Delta x^{+},\Delta z^{+})$ | $(\Delta y^{+}_{\mathrm{max}},\Delta y^{+}_{\mathrm{min}})$ | $Re_{\tau}$
---|---|---|---|---
$(24h,12.8h)$ | $(512,96,512)$ | $(11.8,6.31)$ | $(6.16,0.268)$ | $126.1$

###  II.2 Spectral Analysis on 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. The transport equations
of the temperature-related spectra are derived similarly to those of the
Reynolds stress spectra. In this section, the derivation of the transport
equations of the Reynolds stress spectra is briefly reviewed and that for the
temperature-related turbulence statistics are defined. In the following, the
instantaneous velocities in $x$-, $y$\- and $z$-directions are denoted as
$\tilde{u}=U+u$, $\tilde{v}=v$, and $\tilde{w}=w$, and the instantaneous
temperature is also decomposed as $\tilde{\theta}=\varTheta+\theta$. The
uppercase letters $U=\left\langle\tilde{u}\right\rangle$ and
$\varTheta=\left\langle\tilde{\theta}\right\rangle$ are the mean streamwise
velocity and temperature (here, $\left\langle\right\rangle$ stands for
averaging in $x$\- and $z$-directions and in time), respectively, and the
lowercase letters without the tilde represent the fluctuations around the mean
values. Note here
$\left\langle\tilde{v}\right\rangle=\left\langle\tilde{w}\right\rangle=0$.

####  II.2.1 Transport of Reynolds stresses

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

$\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle
u_{i}u_{j}\right\rangle=P_{ij}-\varepsilon_{ij}+\Phi_{ij}+D^{\nu}_{ij}+D^{t}_{ij},$
(5a) where the terms on the right-hand side are the production ($P_{ij}$), the
viscous dissipation ($\varepsilon_{ij}$), the pressure work ($\Phi_{ij}$), the
viscous diffusion ($D^{\nu}_{ij}$), and the turbulent diffusion ($D^{t}_{ij}$)
terms, which are defined as $\displaystyle P_{ij}$
$\displaystyle=-\left\langle u_{i}u_{k}\right\rangle\frac{\partial
U_{j}}{\partial x_{k}}-\left\langle u_{j}u_{k}\right\rangle\frac{\partial
U_{i}}{\partial x_{k}},$ (5b) $\displaystyle\varepsilon_{ij}$
$\displaystyle=\frac{2}{Re_{\mathrm{w}}}\left\langle\frac{\partial
u_{i}}{\partial x_{k}}\frac{\partial u_{j}}{\partial x_{k}}\right\rangle,$
(5c) $\displaystyle\Phi_{ij}$ $\displaystyle=-\left\langle\frac{\partial
p}{\partial x_{i}}u_{j}\right\rangle-\left\langle\frac{\partial p}{\partial
x_{j}}u_{i}\right\rangle,$ (5d) $\displaystyle D^{\nu}_{ij}$
$\displaystyle=\frac{1}{Re_{\mathrm{w}}}\frac{\partial^{2}\left\langle
u_{i}u_{j}\right\rangle}{\partial x_{k}^{2}},$ (5e) $\displaystyle D^{t}_{ij}$
$\displaystyle=-\frac{\partial\left\langle
u_{i}u_{j}u_{k}\right\rangle}{\partial x_{k}}.$ (5f)

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

$\displaystyle\Phi_{ij}$ $\displaystyle=\Pi_{ij}+D^{p}_{ij},$ (6a)
$\displaystyle\mathrm{where}\quad\Pi_{ij}$ $\displaystyle=\left\langle
p\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial
x_{i}}\right)\right\rangle,$ (6b) $\displaystyle D^{p}_{ij}$
$\displaystyle=-\frac{\partial}{\partial x_{k}}\left(\left\langle
u_{j}p\right\rangle\delta_{ik}+\left\langle
u_{i}p\right\rangle\delta_{jk}\right).$ (6c)

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

Now, we consider a decomposition of the fluctuating components into their
large- and small-scale parts

$\displaystyle{\bf u}={\bf u}^{L}+{\bf
u}^{S},\quad\theta=\theta^{L}+\theta^{S},$ (7)

where the cross correlations between the large- and small-scale part of any
quantities are zero:

$\displaystyle\left\langle u_{i}^{L}u_{j}^{S}\right\rangle=\left\langle
u_{j}^{L}u_{i}^{S}\right\rangle=0,$
$\displaystyle\left\langle\theta^{L}\theta^{S}\right\rangle=\left\langle
u_{i}^{L}\theta^{S}\right\rangle=\left\langle
u_{i}^{S}\theta^{L}\right\rangle=0.$

Such a decomposition is possible based on orthogonal mode decompositions such
as the Fourier mode decomposition and the proper-orthogonal-mode
decomposition. We adopt a decomposition based on the streamwise or spanwise
Fourier modes at a certain cutoff wavenumber $k_{c}$. By the decomposition the
Reynolds stress $\left\langle u_{i}u_{j}\right\rangle$ is decomposed simply
into their large- and small-scale parts as

$\displaystyle\left\langle u_{i}u_{j}\right\rangle=\left\langle
u_{i}^{L}u_{j}^{L}\right\rangle+\left\langle u_{i}^{S}u_{j}^{S}\right\rangle.$
(8)

With a similar manner to the “full” Reynolds stress equation (5), one can
obtain the transport equations of the large- and small-scale parts of
$\left\langle u_{i}u_{j}\right\rangle$ (the details of the derivation are
found in Refs. Kawata and Alfredsson, 2019; Kawata and Tsukahara, 2021) as

$\displaystyle\frac{{\rm d}\left\langle u_{i}^{L}u_{j}^{L}\right\rangle}{{\rm
d}t}=P^{L}_{ij}-\varepsilon^{L}_{ij}+\Phi^{L}_{ij}+D^{\nu,L}_{ij}+D^{t,L}_{ij}-Tr_{ij},$
(9a) $\displaystyle\frac{{\rm d}\left\langle
u_{i}^{S}u_{j}^{S}\right\rangle}{{\rm
d}t}=P^{S}_{ij}-\varepsilon^{S}_{ij}+\Phi^{S}_{ij}+D^{\nu,S}_{ij}+D^{t,S}_{ij}+Tr_{ij}.$
(9b)

Here, ${\rm d}/{\rm d}t=\partial/\partial t+U_{k}\partial/\partial x_{k}$, and
the terms on the right-hand sides of Eqs. (9a, b) are the large- and small-
scale parts of their counterparts in Eq. (5), except for $Tr_{ij}$. In
particular, the first four terms in Eq. (5) (i.e., the production, the viscous
dissipation, the pressure work, and the viscous diffusion) are simply
decomposed into their large- and small-scale parts similarly in Eq. (8): those
in Eq. (9a) are defined as in Eq. (5b–e) with $u_{i}$ and $u_{j}$ replaced by
$u_{i}^{L}$ and $u_{j}^{L}$ (for those in Eq. (9b), replaced by $u_{i}^{S}$
and $u_{j}^{S}$).

On the other hand, the turbulent diffusion is decomposed as

$\displaystyle D^{t,L}=-\frac{\partial\left\langle
u_{i}u_{j}u_{k}\right\rangle^{L}}{\partial x_{k}},\quad
D^{t,S}=-\frac{\partial\left\langle u_{i}u_{j}u_{k}\right\rangle^{S}}{\partial
x_{k}},$ (10)

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 the large- and small-scale part of the
triple velocity correlation $\left\langle u_{i}u_{j}u_{k}\right\rangle$
defined as

$\displaystyle\left\langle u_{i}u_{j}u_{k}\right\rangle^{L}=\left\langle
u_{i}^{L}u_{j}^{L}u_{k}\right\rangle+\left\langle
u_{i}^{S}u_{j}^{L}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{L}u_{j}^{S}u_{k}^{S}\right\rangle,$ (11a) $\displaystyle\left\langle
u_{i}u_{j}u_{k}\right\rangle^{S}=\left\langle
u_{i}^{S}u_{j}^{S}u_{k}\right\rangle+\left\langle
u_{i}^{L}u_{j}^{S}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{S}u_{j}^{L}u_{k}^{L}\right\rangle.$ (11b)

The last term in Eqs. (9a, b), $Tr_{ij}$, is defined as

$\displaystyle Tr_{ij}=-\left\langle u_{i}^{S}u_{k}^{S}\frac{\partial
u_{j}^{L}}{\partial x_{k}}\right\rangle-\left\langle
u_{j}^{S}u_{k}^{S}\frac{\partial u_{i}^{L}}{\partial x_{k}}\right\rangle$
$\displaystyle+\left\langle u_{i}^{L}u_{k}^{L}\frac{\partial
u_{j}^{S}}{\partial x_{k}}\right\rangle+\left\langle
u_{j}^{L}u_{k}^{L}\frac{\partial u_{i}^{S}}{\partial x_{k}}\right\rangle,$
(12)

and as can be seen from Eqs.(9a, b) this term appears in both equations with
opposite signs, which means that $Tr_{ij}$ represents the energy exchange
between the large- and small-scale velocity fields across the cutoff
wavenumber $k_{c}$.

It is worth pointing out here that the turbulent transport terms
($D^{t,L}_{ij}$ and $D^{t,S}_{ij}$) and the interscale energy flux term
($Tr_{ij}$) includes both the large- and small-scale part of fluctuating
velocities, which is in contrast to the other terms in the the transport
equations consisting of only either ${\bf u}^{L}$ or ${\bf u}^{S}$. This is
because both the turbulent transport and the interscale energy flux terms are
the third-order moments of fluctuating velocities (and the velocity
gradients), while the other terms are of the second moments. As
$D^{t,L}_{ij}$, $D^{t,S}_{ij}$, and $Tr_{ij}$ include both large- and small-
scale parts of the velocity field, these terms can be interpreted to represent
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 ($Tr_{ij}$) 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 Reynolds stress spectra $E_{ij}$ are
related as

$\displaystyle E_{ij}=\frac{\partial\left\langle
u_{i}^{L}u^{L}_{j}\right\rangle}{\partial
k_{c}}\left(=-\frac{\partial\left\langle
u_{i}^{S}u_{j}^{S}\right\rangle}{\partial k_{c}}\right),$ (13)

the transport equations of the Reynolds-stress spectra can, therefore, be
derived by differentiating both sides of Eq. (9a) with respect to $k_{c}$ as

$\displaystyle\frac{{\rm d}E_{ij}}{{\rm
d}t}=pr_{ij}-\xi_{ij}+\phi_{ij}+d^{\nu}_{ij}+d^{t}_{ij}+tr_{ij},$ (14)

where the terms on the right-hand side are the $k_{c}$-derivatives of the
counterparts in Eq. (9a). The first five terms represent the spectra of the
corresponding terms of Eq.( 5a): the spectra of the production ($pr_{ij}$),
the viscous dissipation ($\xi_{ij}$), the pressure-work ($\phi_{ij}$), the
viscous diffusion $d^{\nu}_{ij}$, and the turbulent transport $d^{t}_{ij}$.
The interscale transport term $tr_{ij}$ represents the energy gain/loss by the
interscale flux $Tr_{ij}$ at each wavenumebr. In particular, the spectra of
the production ($pr_{ij}$) and the viscous diffusion ($d^{\nu}_{ij}$) are
expressed based on the Reynolds stress spectra as

$\displaystyle pr_{ij}$ $\displaystyle=-E_{ik}\frac{\partial U_{j}}{\partial
x_{k}}-E_{jk}\frac{\partial U_{i}}{\partial x_{k}},$ (15a) $\displaystyle
d^{\nu}_{ij}$
$\displaystyle=\frac{1}{Re_{\mathrm{w}}}\frac{\partial^{2}E_{ij}}{\partial
x_{k}^{2}}.$ (15b)

According to Eq. (6), the pressure-work spectrum $\phi_{ij}$ may be decomposed
into the spectra of $\Pi_{ij}$ and $D^{p}_{ij}$ as

$\displaystyle\phi_{ij}=\pi_{ij}+d^{p}_{ij}.$ (16)

Here, $\pi_{ij}$ is the cospectrum between the pressure and the strain rate,
representing the inter-component energy transfers between different components
of velocity fluctuation at each scale, and $d^{p}_{ij}$ consists of the
velocity-pressure cospectra, representing spatial energy transport by pressure
fluctuation at each scale.

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

$\displaystyle d^{t}_{ij}=-\frac{\partial E_{ijk}}{\partial x_{k}},\quad\quad
tr_{ij}=-\frac{\partial Tr_{ij}}{\partial k_{c}}.$ (17a) Here, $E_{ijk}$ is
the spectra of triple velocity correlations $\left\langle
u_{i}u_{j}u_{k}\right\rangle$ defined as $\displaystyle
E_{ijk}=\frac{\partial\left\langle u_{i}u_{j}u_{k}\right\rangle^{L}}{\partial
k_{c}}\left(=-\frac{\partial\left\langle
u_{i}u_{j}u_{k}\right\rangle^{S}}{\partial k_{c}}\right),$ (17b)

similarly to the definition of the Reynolds stress spectra, Eq. (13). 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 $E_{ijk}$ represent the spatial flux of the Reynolds stress
$\left\langle u_{i}u_{j}\right\rangle$ in the $x_{k}$-direction at each
wavenumber. Again, note that the spatial transport represented by $E_{ijk}$ is
caused by interactions between different scales as indicated by Eqs. (11).

####  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 (5) as

$\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle\theta^{2}\right\rangle=P_{\theta\theta}-\varepsilon_{\theta\theta}+D^{\nu}_{\theta\theta}+D^{t}_{\theta\theta},$
(18a) with the production ($P_{\theta\theta}$), the viscous dissipation
($\varepsilon_{\theta\theta}$), the viscous diffusion
($D^{\nu}_{\theta\theta}$), and the turbulent transport
($D^{t}_{\theta\theta}$) terms defined as $\displaystyle P_{\theta\theta}$
$\displaystyle=-2\left\langle\theta
u_{k}\right\rangle\frac{\partial\varTheta}{\partial x_{k}},$ (18b)
$\displaystyle\varepsilon_{\theta\theta}$
$\displaystyle=\frac{2}{Re_{\mathrm{w}}Pr}\left\langle\frac{\partial\theta}{\partial
x_{k}}\frac{\partial\theta}{\partial x_{k}}\right\rangle,$ (18c)
$\displaystyle D^{\nu}_{ij}$
$\displaystyle=\frac{1}{Re_{\mathrm{w}}Pr}\frac{\partial^{2}\left\langle\theta^{2}\right\rangle}{\partial
x_{k}^{2}},$ (18d) $\displaystyle D^{t}_{\theta\theta}$
$\displaystyle=-\frac{\partial\left\langle\theta^{2}u_{k}\right\rangle}{\partial
x_{k}}.$ (18e)

It is noteworthy here that any term corresponding to $\Phi_{ij}$ in Eq. (5a)
does not exist in Eq. (18a), since the energy equation (2) does not have
pressure-related term.

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

$\left(\frac{\partial}{\partial t}+U_{k}\frac{\partial}{\partial
x_{k}}\right)\left\langle
u_{i}\theta\right\rangle=P_{i\theta}-\varepsilon_{i\theta}+\Phi_{i\theta}+D^{\nu}_{i\theta}+D^{t}_{i\theta},$
(19a) with the production ($P_{i\theta}$), the viscous dissipation
($\varepsilon_{i\theta}$), the pressure-gradient-temperature correlations
($\Phi_{i\theta}$), the viscous diffusion ($D^{\nu}_{i\theta}$) and the
turbulent transport ($D^{t}_{i\theta}$) terms: $\displaystyle P_{i\theta}$
$\displaystyle=-\left\langle u_{k}\theta\right\rangle\frac{\partial
U_{i}}{\partial x_{k}}-\left\langle
u_{i}u_{k}\right\rangle\frac{\partial\varTheta}{\partial x_{k}},$ (19b)
$\displaystyle\varepsilon_{i\theta}$
$\displaystyle=\frac{1}{Re_{\mathrm{w}}}\left(1+\frac{1}{Pr}\right)\left\langle\frac{\partial
u_{i}}{\partial x_{k}}\frac{\partial\theta}{\partial x_{k}}\right\rangle,$
(19c) $\displaystyle\Phi_{i\theta}$ $\displaystyle=-\left\langle\frac{\partial
p}{\partial x_{i}}\theta\right\rangle,$ (19d) $\displaystyle
D^{\nu}_{i\theta}$
$\displaystyle=\frac{1}{Re_{\mathrm{w}}}\frac{\partial}{\partial
x_{k}}\left(\left\langle\frac{\partial u_{i}}{\partial
x_{k}}\theta\right\rangle+\frac{1}{Pr}\left\langle\frac{\partial\theta}{\partial
x_{k}}u_{i}\right\rangle\right),$ (19e) $\displaystyle D^{t}_{i\theta}$
$\displaystyle=-\frac{\partial\left\langle u_{i}\theta
u_{k}\right\rangle}{\partial x_{k}}.$ (19f)

Based on the decomposition given by Eq. (7), the temperature fluctuation and
the turbulent heat fluxes are also split into their large- and small-scale
parts, similarly to the Reynolds stress decomposition (8), as

$\displaystyle\left\langle\theta^{2}\right\rangle=\left\langle\theta^{L}\theta^{L}\right\rangle+\left\langle\theta^{S}\theta^{S}\right\rangle,$
(20a) $\displaystyle\left\langle u_{i}\theta\right\rangle=\left\langle
u_{i}^{L}\theta^{L}\right\rangle+\left\langle
u_{i}^{S}\theta^{S}\right\rangle,$ (20b)

and therefore the transport equations of their large- and small-scale parts
can be derived similarly to Eqs. (9). Then, these large- and small-scale parts
of the temperature-related statistics are related to their spectra, similarly
to Eq. (13), as

$\displaystyle
E_{\theta\theta}=\frac{\partial\left\langle\theta^{L}\theta^{L}\right\rangle}{\partial
k_{c}},\quad E_{i\theta}=\frac{\partial\left\langle
u_{i}^{L}\theta^{L}\right\rangle}{\partial k_{c}},$ (21)

and hence 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 $\left\langle
u_{i}^{L}\theta^{L}\right\rangle$. The transport equation of the temperature-
fluctuation spectrum $E_{\theta\theta}$ is expressed as

$\displaystyle\frac{{\rm d}E_{\theta\theta}}{{\rm d}t}$
$\displaystyle=pr_{\theta\theta}-\xi_{\theta\theta}+d^{\nu}_{\theta\theta}+d^{t}_{\theta\theta}+tr_{\theta\theta}$
(22)

with the terms on the right-hand side representing the spectra of the
counterparts in Eq. (18). The interscale and the turbulent spatial transport
terms are defined as

$\displaystyle tr_{\theta\theta}=-\frac{\partial Tr_{\theta\theta}}{\partial
k_{c}},\quad\quad d^{t}_{\theta\theta}=-\frac{\partial E_{\theta\theta
k}}{\partial x_{k}},$ (23a) where $Tr_{\theta\theta}$ and $E_{\theta\theta k}$
are the interscale and the spectral spatial fluxes of the temperature
fluctuation defined as $\displaystyle Tr_{\theta\theta}$
$\displaystyle=-2\left\langle\theta^{S}u_{k}^{S}\frac{\partial\theta^{L}}{\partial
x_{k}}\right\rangle-2\left\langle\theta^{L}u_{k}^{L}\frac{\partial\theta^{S}}{\partial
x_{k}}\right\rangle,$ (23b) $\displaystyle E_{\theta\theta k}$
$\displaystyle=\frac{\partial\left\langle\theta^{2}u_{k}\right\rangle^{L}}{\partial
k_{c}}=-\frac{\partial\left\langle\theta^{2}u_{k}\right\rangle^{S}}{\partial
k_{c}},$ (23c) 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 parts of the spatial flux of temperature fluctuation
$\left\langle\theta^{2}u_{k}\right\rangle$:
$\displaystyle\left\langle\theta^{2}u_{k}\right\rangle^{L}$
$\displaystyle=\left\langle\theta^{L}\theta^{L}u_{k}\right\rangle+2\left\langle\theta^{L}\theta^{S}u_{k}^{S}\right\rangle,$
(23d) $\displaystyle\left\langle\theta^{2}u_{k}\right\rangle^{S}$
$\displaystyle=\left\langle\theta^{S}\theta^{S}u_{k}\right\rangle+2\left\langle\theta^{L}\theta^{S}u_{k}^{L}\right\rangle.$
(23e)

Similarly, the transport equation of the turbulent heat flux spectrum
$E_{i\theta}$ is obtained as

$\displaystyle\frac{{\rm d}E_{i\theta}}{{\rm d}t}$
$\displaystyle=pr_{i\theta}-\xi_{i\theta}+\phi_{i\theta}+d^{\nu}_{i\theta}+d^{t}_{i\theta}+tr_{i\theta},$
(24)

and the terms on the right-hand side (except $tr_{i\theta}$) are, again, the
spectra of their counterparts in Eq. (19). The interscale and the turbulent
spatial transport terms are defined in the same manners as in other spectral
transport equations:

$\displaystyle tr_{i\theta}=-\frac{\partial Tr_{i\theta}}{\partial
k_{c}},\quad\quad d^{t}_{i\theta}=-\frac{\partial E_{i\theta k}}{\partial
x_{k}},$ (25a) where the interscale flux $Tr_{i\theta}$ and the spectral
spatial flux $E_{i\theta k}$ are respectively defined as $\displaystyle
Tr_{i\theta}=$ $\displaystyle-\left\langle\theta^{S}u_{k}^{S}\frac{\partial
u_{i}^{L}}{\partial x_{k}}\right\rangle-\left\langle
u_{i}^{S}u_{k}^{S}\frac{\partial\theta^{L}}{\partial x_{k}}\right\rangle$
$\displaystyle+\left\langle\theta^{L}u_{k}^{L}\frac{\partial
u_{i}^{S}}{\partial x_{k}}\right\rangle+\left\langle
u_{i}^{L}u_{k}^{L}\frac{\partial\theta^{S}}{\partial x_{k}}\right\rangle,$
(25b) $\displaystyle E_{i\theta k}=$ $\displaystyle\frac{\partial\left\langle
u_{i}\theta u_{k}\right\rangle^{L}}{\partial
k_{c}}=-\frac{\partial\left\langle u_{i}\theta
u_{k}\right\rangle^{S}}{\partial k_{c}}.$ (25c) 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 $\displaystyle\left\langle
u_{i}\theta u_{k}\right\rangle^{L}=\left\langle
u_{i}^{L}\theta^{L}u_{k}\right\rangle+\left\langle
u_{i}^{L}\theta^{S}u_{k}^{S}\right\rangle+\left\langle
u_{i}^{S}\theta^{L}u_{k}^{S}\right\rangle,$ (25d) $\displaystyle\left\langle
u_{i}\theta u_{k}\right\rangle^{S}=\left\langle
u_{i}^{S}\theta^{S}u_{k}\right\rangle+\left\langle
u_{i}^{S}\theta^{L}u_{k}^{L}\right\rangle+\left\langle
u_{i}^{L}\theta^{S}u_{k}^{L}\right\rangle.$ (25e)

##  III Results

###  III.1 Basic Statistics and Spectra of Turbulent Heat and Momentum
Transfers

Figure 1: Profiles of mean streamwise velocity $U$ and mean temperature
$\varTheta$ in (top) outer and (bottom) inner scaling. In the bottom panel,
the black dashed line presents reference data of mean streamwise velocity
obtained in Ref. Kawata and Tsukahara, 2021 with an extremely large
computational domain $(L_{x}^{\ast},L_{z}^{\ast})=(96h,12.8h)$, and the grey
chained lines indicate $U^{+}=y^{+}$ and $U^{+}=2.44\ln y^{+}+5.1$ for mean
velocity profile and $\varTheta^{+}=Pry^{+}$ for mean temperature profile.

Figure 1 presents the profiles of the mean streamwise velocity $U$ and the
mean temperature $\varTheta$ in the outer and inner scalings. Here,
$T^{\ast}_{\tau}$ used for the inner scaling of temperature is the friction
temperature defined as

$\displaystyle
T_{\tau}^{\ast}=\frac{Q_{\mathrm{w}}^{\ast}}{\rho^{\ast}c_{p}^{\ast}u_{\tau}^{\ast}}$
(26)

where $Q_{\mathrm{w}}^{\ast}$ and $c_{p}^{\ast}$ are the mean heat flux on the
wall and the specific heat at constant pressure, respectively. As shown in the
top panel, the profiles of $U$ and $\varTheta$ are quite similar to each
other, but the temperature gradient in the vicinity of the wall is certainly
smaller than the velocity gradient, which is displayed more clearly by the
plot in the inner scaling in the bottom panel. This is attributable to the
Prandtl number being smaller than $1$, which means that the molecular
diffusion effect is more significant in the temperature field. In the bottom
panel the profile of $U^{+}$ obtained in our previous studyKawata and
Tsukahara (2021) with an extremely large computational domain
($L_{x}^{\ast}=96.0h$, $L_{z}^{\ast}=12.8h$) is also given by the black dashed
line for reference. As can be seen here, the profile obtained in the present
computation is in a good agreement with the reference data.

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, i.e., the streamwise and wall-normal turbulent heat fluxes
$\left\langle u\theta\right\rangle$ and $\left\langle v\theta\right\rangle$.
Note that in the figure only the lower half of the channel $0\ll y\ll 0.5$ is
shown as the statistics are averaged between the upper and lower halves of the
channel. Also in this figure the reference data from our previous work Kawata
and Tsukahara (2021) are shown (black circles) and the consistency between the
present and reference data can be confirmed, indicating that no effect of the
finite domain size in the present computation. It is also shown in Fig. 2 that
the temperature fluctuation $\left\langle\theta^{2}\right\rangle$ and the
streamwise-velocity-temperature correlation $\left\langle
u\theta\right\rangle$ give similar profiles to the streamwise velocity
fluctuation $\left\langle u^{2}\right\rangle$, which can be attributed to the
similar profiles of the mean streamwise velocity $U$ and the mean temperature
$\varTheta$, as shown in Fig. 1. The smaller peak magnitudes of the
$\left\langle\theta^{2}\right\rangle$ profile may correspond to the fact that
the mean temperature gradient is certainly smaller than the mean velocity
gradient in the near-wall region. It is also observed that the peak magnitude
of the $\left\langle u\theta\right\rangle$ is as large as those of
$\left\langle u^{2}\right\rangle$ and $\left\langle\theta^{2}\right\rangle$,
which indicates the strong cross correlation between $u$ and $\theta$.

The correlation $\left\langle v\theta\right\rangle$ is a particularly
important temperature-related quantity, as it represents the wall-normal heat
transfer by turbulent fluid motions, similarly to the Reynolds shear stress
$\left\langle uv\right\rangle$ representing the momentum transfer by
turbulence. As shown in Fig. 2, the profiles of $\left\langle
v\theta\right\rangle$ and $\left\langle uv\right\rangle$ are almost on top on
each other, which indicates a strong similarity between the turbulent momentum
and the heat transfers.

Figure 2: Profiles of (solid lines) the Reynolds stresses $\left\langle
u_{i}u_{j}\right\rangle$ and (dashed lines) 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,
$u_{\tau}^{\ast}$ and/or $T_{\tau}^{\ast}$, and only $0\ll y\ll 0.5$ is shown.
The black circles indicate the profiles of the Reynolds stresses obtained with
an extremely large computational domain
$(L_{x}^{\ast},L_{z}^{\ast})=(96h,12.8h)$Kawata and Tsukahara (2021).

Figure 3: Space-wavelength ($y$-$\lambda_{x}$) diagrams of premultiplied
streamwise one-dimensional spectra of (a) streamwise turbulent energy
$k_{x}E^{x}_{uu}$, turbulent kinetic energy
$k_{x}E^{x}_{kt}=k_{x}(E^{x}_{uu}+E^{x}_{vv}+E^{x}_{ww})/2$, and (c)
temperature fluctuation $k_{x}E^{x}_{\theta\theta}$. The values are scaled by
(a,b) $u_{\tau}^{\ast 2}$ and (c) $T_{\tau}^{\ast 2}$.

Figure 4: Distributions of spanwise one-dimensional spectra of (a) streamwise
turbulent energy $E^{z}_{uu}$, (b) turbulent kinetic energy $E^{z}_{kt}$, and
(c) temperature fluctuation $E^{z}_{\theta\theta}$, presented in the same
manner as in Fig. 4.

Next, we focus on the spectral quantities of the turbulent statistics. Figure
4 presents the space-wavelength ($y$-$\lambda_{x}$) diagrams of premultiplied
one-dimensional streamwise energy spectra, comparing those of the streamwise
turbulent energy $E^{x}_{uu}$, the turbulent kinetic energy $E^{x}_{kt}$
(here, $k_{t}=(\left\langle u^{2}\right\rangle+\left\langle
v^{2}\right\rangle+\left\langle w^{2}\right\rangle)$/2), and the temperature
fluctuation $E^{x}_{\theta\theta}$. Note here that these spectra distributions
are also shown only for the lower half of the channel $0\ll y\ll 0.5$, as they
are averaged between the upper and lower halves of the channel. As shown in
the panel (a), the distribution of $E^{x}_{uu}$ indicates an energy peak in
the near-wall region at relatively small streamwise wavelength
$\lambda_{x}^{+}\approx 600$, which corresponds to the near-wall coherent
structures. It is also observed that the spectral energy increases at the
largest wavelength $\lambda_{x}=24$ throughout the channel, representing the
very-large-scale structure.

The energy spectrum of the temperature fluctuation $E^{x}_{\theta\theta}$
given in Fig. 4(c) presents a qualitatively similar distribution to that of
$E^{x}_{uu}$, showing the energy peaks at relatively small wavelength near the
wall and at the largest wavelength $\lambda_{x}=24$ in the channel-core
region. This observation is consistent with the fact that the profiles of
$\left\langle u^{2}\right\rangle$ and $\left\langle\theta^{2}\right\rangle$
are similar to each other. One can, however, also find that the distributions
of the temperature-fluctuation spectrum $E^{x}_{\theta\theta}$ spans to
smaller wavelengths than that of the streamwise turbulent energy $E^{x}_{uu}$:
while in the $E^{x}_{uu}$ distribution the energy larger than 30% of the peak
magnitude is distributed roughly in the range $\lambda_{x}>0.5$, the energy
with the same magnitude spans down to about $\lambda_{x}=0.2$ in the
$E^{x}_{\theta\theta}$ distribution. The difference between the distributions
of $E^{x}_{uu}$ and $E^{x}_{\theta\theta}$ is found particularly clearly in
the channel-core region, as the energy peak of $E^{x}_{uu}$ is located at
around $\lambda_{x}\approx 3$, while that of the $E^{x}_{\theta\theta}$
distribution is at $\lambda_{x}\approx 1$. This observation is consistent with
the report by Hane et al.Hane  _et al._ (2006).

The difference between the distributions of $E^{x}_{uu}$ and
$E^{x}_{\theta\theta}$ is partly attributable to the energy redistribution by
the pressure-strain correlation, as it absorbs a certain amount of energy from
$E^{x}_{uu}$ and transfers it to other components $E^{x}_{vv}$ and
$E^{x}_{ww}$ at the relatively small wavelengthsLee and Moser (2019); Kawata
and Tsukahara (2021). Figure 4(b) presents the turbulent kinetic energy
spectrum $E^{x}_{kt}=(E^{x}_{uu}+E^{x}_{vv}+E^{x}_{ww})/2$, and it is seen
here that the distribution of $E^{x}_{kt}$ is slightly shifted to small-
wavelength side compared to that of $E^{x}_{uu}$ alone, because of the
spectral energy at relatively small $\lambda_{x}$ possessed by $E^{x}_{vv}$
and $E^{x}_{ww}$. However, the temperature fluctuation spectrum
$E^{x}_{\theta\theta}$ is still distributed in somewhat smaller wavelength
range than $E^{x}_{kt}$, compare Figs. 4(b) and (c).

This shift of the distribution of $E^{x}_{\theta\theta}$ towards small
$\lambda_{x}$ side in comparison with the $E^{x}_{kt}$ distribution might be a
somewhat surprising outcome, because the effect of molecular diffusion is more
significant in the temperature field than in the velocity field as the Prandtl
number is smaller than 1. Hence, one might expect that small-scale
fluctuations would be damped in the fluctuating temperature field as compared
to the velocity field. It has, however, turned out that small-scale
fluctuations are more energetic in the temperature field than in the velocity
field, as observed in Fig. 4.

Such difference between the spectra of the velocity and temperature
fluctuations is not observed from the spanwise spectra. Figure 4 presents the
distributions of the spanwise one-dimensional spectra $E^{z}_{uu}$,
$E^{z}_{kt}$, and $E^{z}_{\theta\theta}$ in the same manner as in Fig. 4. As
shown in the panel (a), the distribution of the streamwise turbulent energy
spectrum $E^{z}_{uu}$ shows a near-wall energy peak at $\lambda_{z}^{+}\approx
100$ and a significant energy band across the channel at $\lambda_{z}\approx
2.5$, which correspond to the near-wall coherent and the very-large-scale
structures, respectively. The temperature-fluctuation spectrum
$E^{z}_{\theta\theta}$ presents a similar distribution to $E^{z}_{uu}$ in
terms of the peak wavelengths, as shown in Fig. 4(b). As observed here, the
spectral energies $E^{z}_{uu}$ and $E^{z}_{\theta\theta}$ are distributed in
the same $\lambda_{z}$ range, which is in contrast to $E^{x}_{uu}$ and
$E^{x}_{\theta\theta}$ as a function of the streamwise wavelength
$\lambda_{x}$.

It is also noteworthy here that the turbulent kinetic energy spectrum
$E^{z}_{kt}$ also gives a similar distribution to $E^{z}_{uu}$, as shown in
Fig. 4(b), unlike the streamwise counterparts of $E^{x}_{uu}$ and
$E^{x}_{kt}$. This is presumably because the $\lambda_{z}$ range where the
lateral turbulent energies $E^{z}_{vv}$ and $E^{z}_{ww}$ are distributed are
not so small as compared to the streamwise turbulent energy $E^{z}_{uu}$.

Figure 5: Distributions of streamwise one-dimensional cospectra of (a) the
Reynolds shear stress $k_{x}E^{x}_{-uv}$ and (b) turbulent heat flux
$k_{x}E^{x}_{-v\theta}$, presented in the same manner as in Fig. 4. The values
are scaled by $u_{\tau}^{\ast 2}$ and $u_{\tau}^{\ast}T_{\tau}^{\ast}$,
respectively.

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

Similar tendencies are also observed by the spectra of the turbulent momentum
and heat fluxes. Figures 6 and 6 present the distributions of one-dimensional
streamwise and spanwise cospectra of turbulent fluxes, respectively, comparing
those of the Reynolds shear stress and the turbulent heat flux. Note here that
as $-\left\langle uv\right\rangle$ and $-\left\langle v\theta\right\rangle$
represent the wall-normal transports of momentum and heat by turbulent fluid
motions, respectively, their cospectra given in Figs. 6 and 6 represent the
turbulent momentum and heat transfers at each length scale. As shown in Fig.
6, the distributions of the streamwise spectra of the Reynolds shear stress
$E^{x}_{-uv}$ and the turbulent heat flux $E^{x}_{-v\theta}$ are qualitatively
similar, but the distribution of $E^{x}_{-v\theta}$ is somewhat shifted
towards the smaller-wavelength side as compared to $E^{x}_{-uv}$ particularly
in the channel-core region. The spanwise spectra $E^{z}_{-uv}$ and
$E^{z}_{-v\theta}$, on the other hand, give quite similar distributions as
presented in Fig. 6, and the difference between the spanwise length scales are
not observed, consistently to the tendencies shown by the turbulent energy
spectra in Figs. 4 and 4. These observations indicate that the turbulent
momentum and heat transfers are certainly similar scale-by-scale, but the
turbulent heat flux tends to occur at smaller streamwise length scales.

The Reynolds shear stress $-\left\langle uv\right\rangle$ and the turbulent
heat flux $-\left\langle v\theta\right\rangle$ are closely related to the wall
shear stress $\tau_{\mathrm{w}}$ and the wall heat flux $q_{\mathrm{w}}$,
respectively. Integrating the averaged momentum and energy transport equations
across the chanel, one obtains

$\displaystyle\tau_{\mathrm{w}}$
$\displaystyle=\underbrace{\frac{1}{Re_{\mathrm{w}}}}_{\tau_{\mathrm{w,lam}}}+\underbrace{\int^{1}_{0}(-\left\langle
uv\right\rangle)\mathrm{d}y}_{\Delta\tau_{\mathrm{w}}},$ (27a) $\displaystyle
q_{\mathrm{w}}$
$\displaystyle=\underbrace{\frac{1}{PrRe_{\mathrm{w}}}}_{q_{\mathrm{w,lam}}}+\underbrace{\int^{1}_{0}(-\left\langle
v\theta\right\rangle)\mathrm{d}y}_{\Delta q_{\mathrm{w}}},$ (27b)

where the first terms of $\tau_{\mathrm{w,lam}}$ and $q_{\mathrm{w,lam}}$ are
the laminar values of the shear stress and the heat flux on the wall,
respectively, and the second terms represent the increase from them due to
turbulent fluid motions. Equations (27) are the Fukagata-Iwamoto-Kasagi
identity(Fukagata, Iwamoto, and Kasagi, 2002) for the plane Couette
configuration. Taking the streamwise one-dimensional spectra $E^{x}_{-uv}$ and
$E^{x}_{-v\theta}$ for example, $\Delta\tau_{\mathrm{w}}$ and $\Delta
q_{\mathrm{w}}$ are expressed in terms of $E^{x}_{-uv}$ and $E^{x}_{-v\theta}$
as

$\displaystyle\Delta\tau_{\mathrm{w}}$
$\displaystyle=\int^{1}_{0}\int^{\infty}_{0}E^{x}_{-uv}(y,k_{x})\mathrm{d}k_{x}\mathrm{d}y=\int^{\infty}_{0}\mathcal{E}^{x}_{-uv}(k_{x})\mathrm{d}k_{x},$
(28a) $\displaystyle\Delta q_{\mathrm{w}}$
$\displaystyle=\int^{1}_{0}\int^{\infty}_{0}E^{x}_{-v\theta}(y,k_{x})\mathrm{d}k_{x}\mathrm{d}y=\int^{\infty}_{0}\mathcal{E}^{x}_{-v\theta}(k_{x})\mathrm{d}k_{x},$
(28b) where $\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$ are the
spectra of turbulent momentum and heat transfers integrated across the
channel:
$\displaystyle\mathcal{E}^{x}_{-uv}=\int^{1}_{0}E^{x}_{-uv}(y,k_{x})\mathrm{d}y,$
(28c)
$\displaystyle\mathcal{E}^{x}_{-v\theta}=\int^{1}_{0}E^{x}_{-v\theta}(y,k_{x})\mathrm{d}y.$
(28d)

Equations (28) show that the integrated spectra $\mathcal{E}^{x}_{-uv}$ and
$\mathcal{E}^{x}_{-v\theta}$ represent the spectral contents of
$\Delta\tau_{\mathrm{w}}$ and $\Delta q_{\mathrm{w}}$, respectively.
Similarly, one can also similarly define the integrated spanwise spectra,
$\mathcal{E}^{z}_{-uv}(k_{z})$ and $\mathcal{E}^{z}_{-v\theta}(k_{z})$, which
represent the contributions to $\Delta\tau_{\mathrm{w}}$ and $\Delta
q_{\mathrm{w}}$ from each spanwise wavenumber.

Figure 7: Premultiplied spectra of turbulent wall shear stress and turbulent
wall heat flux: (a) streamwise spectra $k_{x}\mathcal{E}^{x}_{-uv}$ and
$k_{x}\mathcal{E}^{x}_{-v\theta}$, (b) spanwise spectra
$k_{z}\mathcal{E}^{z}_{-uv}$ and $k_{z}\mathcal{E}^{z}_{-v\theta}$. The values
are scaled by $u_{\tau}^{\ast 2}h^{\ast}$ and
$u_{\tau}^{\ast}T_{\tau}^{\ast}h^{\ast}$, respectively.

The spectra of the wall shear stress and the wall heat flux are presented in
Fig. 7. In the panel (a), the streamwise one-dimensional spectra of the wall
shear stress $\mathcal{E}^{x}_{-uv}$ and that of the wall heat flux
$\mathcal{E}^{x}_{-v\theta}$ are compared. One can see here that
$\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$ show similar profiles
to each other. Both profiles have a small peak at the largest streamwise
wavelength $\lambda_{x}=24$, which represents the contribution by the very-
large-scale structures to $\Delta\tau_{\mathrm{w}}$ and $\Delta
q_{\mathrm{w}}$, and also show their largest peak at relatively small
wavelengths on the order of $\lambda_{x}\approx 1$. However, the peak of the
$\mathcal{E}^{x}_{-v\theta}$ profile is located at a somewhat smaller
wavelength than that of the $\mathcal{E}^{x}_{-uv}$, which corresponds to the
difference between the distributions of $E^{x}_{-uv}$ and $E^{x}_{-v\theta}$
observed in Fig. 6. This means, again, that the streamwise length scales that
mainly contribute to the wall heat flux is somewhat smaller to those
responsible to the wall shear stress, despite $Pr<1$. The spanwise spectra of
the wall shear stress $\mathcal{E}^{z}_{-uv}$ and the wall heat flux
$\mathcal{E}^{z}_{-v\theta}$, however, does not show such difference in the
scales, as presented in Fig. 7(b). Both profiles of $\mathcal{E}^{z}_{-uv}$
and $\mathcal{E}^{z}_{-v\theta}$ indicate significant peaks at
$\lambda_{z}^{+}\approx 100$ and $\lambda_{z}=2.1$–2.5, corresponding to the
near-wall and very-large-scale structures, respectively, and these peaks of
$\mathcal{E}^{z}_{-uv}$ and $\mathcal{E}^{z}_{-v\theta}$ are located at the
same wavelengths, unlike the peaks observed in the streamwise spectra
$\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$.

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

Now we focus on the interscale and spatial fluxes of the temperature-related
statistics. As shown in the previous section, the streamwise spectra indicate
that the temperature fluctuation is more energetic at small scales than the
velocity fluctuation, although the spanwise spectra do not show any
significant difference. Hence, we focus on the transports of the streamwise
spectra in the following.

Figure 8 compares the distributions of the interscale fluxes (in the
streamwise wavenumber direction) of the turbulent kinetic energy $Tr^{x}_{kt}$
and the temperature fluctuation $Tr^{x}_{\theta\theta}$. It should be noted
here that according to the definition in Eq. (9) the positive values of the
interscale flux represent the transfer from larger to smaller scales. As shown
in Fig. 8(a), the interscale turbulent energy flux $Tr^{x}_{kt}$ basically
indicates forward energy transfers, i.e., from larger to smaller scales.

Figure 8: Space-wavelength ($y$-$\lambda_{x}$) diagrams of streamwise
interscale fluxes of (a) turbulent kinetic energy $Tr^{x}_{kt}$ and (b)
temperature fluctuation $Tr^{x}_{\theta\theta}$. The values are scaled by
$u_{\tau}^{\ast 4}/\nu^{\ast}$ and $u_{\tau}^{\ast 2}T_{\tau}^{\ast
2}/\nu^{\ast}$, respectively. Figure 9: Streamwise interscale fluxes of (a)
the Reynolds shear stress $Tr^{x}_{-uv}$ and (b) turbulent heat flux
$Tr^{x}_{-v\theta}$, presented in the same manner as in Fig. 8.

The distribution of the interscale flux of the temperature fluctuation
$Tr^{x}_{\theta\theta}$ is similar to that of the $Tr^{x}_{kt}$ distribution
as presented in Fig. 8(b), but it appears to be somewhat shifted towards the
smaller-wavelength side. The difference is particularly visible in the central
region of the channel. This tendency of the $Tr^{x}_{\theta\theta}$
distribution is consistent with the observation in Fig. 4 that the
distribution of $E^{x}_{\theta\theta}$ spans to smaller wavelengths than the
$E^{x}_{kt}$ distribution.

Figure 9 compares the distributions of the streamwise interscale fluxes of the
Reynolds shear stress $Tr^{x}_{-uv}$ and the temperature-velocity correlation
$Tr^{x}_{-v\theta}$. As shown, the distributions of $Tr^{x}_{-uv}$ and
$Tr^{x}_{-v\theta}$ are similar to each other, similarly to the interscale
fluxes of turbulent energies in Fig. 8. Both $Tr^{x}_{-uv}$ and
$Tr^{x}_{-v\theta}$ basically present forward interscale transfers and their
maxima are found at similar locations in the diagrams. It is shown, again,
that the distribution of $Tr^{x}_{-v\theta}$ spreads to smaller wavelengths
than the Reynolds shear stress flux $Tr^{x}_{-uv}$, which is consistent with
the corresponding cospectrum distributions given in Fig. 6.

Next we investigate the spectra of turbulent spatial fluxes of the turbulent
energies and the temperature-related statistics. Figure 10 presents the
distributions of the streamwise one-dimensional spectrum of the turbulent
spatial transport of the kinetic energy
$E^{x}_{iiv}/2=(E^{x}_{uuv}+E^{x}_{vvv}+E^{x}_{wwv})/2$ and that of the
temperature fluctuation $E^{x}_{\theta\theta v}$, which represent the spatial
transports of $k_{t}$ and $\left\langle\theta^{2}\right\rangle$ in
$y$-direction at each wavelength $\lambda_{x}$, respectively: see also Eqs.
(17b) and (23c) for their definitions. As shown in the panel (a), the spatial
energy transport spectrum $E^{x}_{iiv}/2$ indicates a broad energy
distribution over a wide range of $\lambda_{x}$. The region of negative
$E^{x}_{iiv}/2$ (where the turbulent kinetic energy is transported towards the
wall) in the near-wall region spans from the largest wavelength
($\lambda_{x}=24.0$) to the middle wavelength range around $\lambda_{x}=1.0$
($\lambda_{x}\approx 250$), while the positive region ($k_{t}$ is transported
away from the bottom wall) spreads down to further smaller wavelengths around
$\lambda_{x}=0.2$ ($\lambda_{x}^{+}\approx 50$).

Figure 10: Space-wavelength ($y$-$\lambda_{x}$) diagram of premultiplied
streamwise one-dimensional spectra of turbulent spatial fluxes: (a) turbulent
kinetic energy transport
$k_{x}E^{x}_{iiv}/2=k_{x}(E^{x}_{uuv}+E^{x}_{vvv}+E^{x}_{wwv})/2$ and (b)
temperature fluctuation transport $k_{x}E^{x}_{\theta\theta v}$. The values
are scaled by $u_{\tau}^{\ast 3}$ and $u_{\tau}^{\ast}T_{\tau}^{\ast 2}$,
respectively.

The distribution of the $\left\langle\theta^{2}\right\rangle$ transport
spectrum, $E^{x}_{\theta\theta v}$, also presents contributions from a wide
range of the streamwise wavelength $\lambda_{x}$ similarly to the
$E^{x}_{iiv}/2$ distribution, as shown in Fig. 10(b). However, in the
$E^{x}_{\theta\theta v}$ distribution, the negative spatial transport in the
near-wall region is shown to occur at much smaller wavelengths than in the
$E^{x}_{iiv}/2$ spectrum distribution, and there is a secondary peak of the
positive spectral transport at small wavelength around $\lambda_{x}\approx
0.3$ ($\lambda^{+}\approx 60$). This secondary peak accounts for about 13% of
the maximum level of the overall turbulent spatial transport of the
temperature fluctuation $\left\langle\theta^{2}v\right\rangle$, while the
turbulent energy transport spectrum $E^{x}_{iiv}/2$ in the same $\lambda_{x}$
range accounts for about 7% of $(\left\langle u^{2}v\right\rangle+\left\langle
v^{3}\right\rangle+\left\langle w^{2}v\right\rangle)/2$. This difference
indicates that the turbulent spatial transport of the temperature fluctuation
is more significant at relatively small wavelengths compared to the turbulent
kinetic energy transport.

Figure 11: Streamwise one-dimensional spectra of turbulent spatial fluxes of
(a) the Reynolds shear stress $k_{x}E^{x}_{-uvv}$ and (b) turbulent heat flux
$k_{x}E^{x}_{-v\theta v}$. The values are scaled by $u_{\tau}^{\ast 3}$ and
$u_{\tau}^{\ast 2}T_{\tau}^{\ast}$, respectively.

The spectral spatial fluxes of the Reynolds shear stress $E^{x}_{-uvv}$ and
the turbulent heat flux $E^{x}_{-v\theta v}$ are also presented in Fig. 11. As
shown in the panel (a), the distribution of $E^{x}_{-uvv}$ indicates the wall-
ward (negative) transports in the near-wall region and the in-ward (positive)
transports in the channel core for a wide range of the wavelength similarly to
the turbulent kinetic energy flux $E^{x}_{iiv}/2$. It is also seen by
comparing the distributions of $E^{x}_{-uvv}$ (Fig. 11a) and $E^{x}_{iiv}/2$
(Fig. 10a) that the Reynolds shear stress transport occurs at relatively large
wavelength compared to the turbulent kinetic energy transport. The spectrum of
the turbulent heat flux transport $E^{x}_{-v\theta v}$ also present a similar
distribution, as shown in the panel (b), where the negative and positive
transports are indicated in the near- and far-wall regions, respectively, at
relatively large wavelengths. It is also seen that the negative transports
take place in a wide range of the channel at relatively small wavelengths
$\lambda_{x}\approx 0.3$. Such spatial transports at small wavelength are also
observed by the temperature-fluctuation flux $E^{x}_{\theta\theta v}$ as shown
in Fig. 10(b), which is presumably related with the fact that the temperature
fluctuation at small streamwise length scale is more energetic than the
velocity fluctuation.

As described so far, the spectral distributions of the fluctuating velocity
and temperature fields have been investigated including their interscale and
spatial transports, and it has been observed that the temperature-related
statistics basically show similar distributions to the corresponding
turbulence statistics, which indicates a close similarity between the
fluctuating velocity and temperature fields. Some differences have, however,
also been found. A particularly interesting observation is that the
temperature fluctuation at small streamwise length scales is more energetic
than those in the fluctuating velocity field, although $Pr<1$ and therefore
the effect of molecular diffusion is greater in the temperature field. This
point will be discussed in detail in the next section.

##  IV Discussion

###  IV.1 Spectral Analysis on Transport of Turbulent Kinetic Energy and
Temperature Fluctuation

Now we investigate the budget balance of the spectral transport equations of
the temperature-related statistics, in order to address the difference in the
dominating streamwise length scales of the heat and momentum transfers
observed in the previous section. First, we focus on the spectral transport of
the turbulent kinetic energy and the temperature fluctuation.

Figure 12: Space-wavelength ($y$-$\lambda_{x}$) diagrams of premultiplied
streamwise one-dimensional spectra of (a) turbulent kinetic energy production
$k_{x}pr^{x}_{uu}$ and (b) temperature fluctuation production
$k_{x}pr^{x}_{\theta\theta}$. The values are scaled by $u_{\tau}^{\ast
4}/\nu^{\ast}$ and $u_{\tau}^{\ast 2}T_{\tau}^{\ast 2}/\nu^{\ast}$,
respectively.

Figure 12 compares the production spectra of the turbulent kinetic energy
$pr^{x}_{kt}$ and the temperature fluctuation $pr^{x}_{\theta\theta}$. These
production spectra are directly related to the Reynolds shear stress
cospectrum $E^{x}_{-uv}$ and the turbulent heat flux spectrum
$E^{x}_{-v\theta}$ as

$\displaystyle pr^{x}_{kt}$
$\displaystyle=E^{x}_{-uv}\frac{\mathrm{d}U}{\mathrm{d}y},$ (29a)
$\displaystyle pr^{x}_{\theta\theta}$
$\displaystyle=2E^{x}_{-v\theta}\frac{\mathrm{d}\varTheta}{\mathrm{d}y},$
(29b)

in the present flow configuration, and therefore the $\lambda_{x}$ ranges
where the productions of $k_{t}$ and $\left\langle\theta^{2}\right\rangle$
occur are determined by the distributions of $E^{x}_{-uv}$ and
$E^{x}_{-v\theta}$, respectively. As shown in Fig. 12, these production
spectra present similar distributions to each other, presenting a near-wall
peak located at similar locations $y^{+}\approx 16$ and $\lambda_{x}\approx
0.2$ ($\lambda_{x}^{+}\approx 500$). However, corresponding to the
distributions of $E^{x}_{-uv}$ and $E^{x}_{-v\theta}$ given in Fig. 6, the
distribution of the temperature-fluctuation production $pr^{x}_{\theta\theta}$
is shown to spread down to smaller wavelengths than the turbulent energy
production $pr^{x}_{kt}$. The difference between the distributions of these
production spectra is, however, not as distinct as that between the
$E^{x}_{kt}$ and $E^{x}_{\theta\theta}$ distributions observed in Fig. 4.

Figure 13: Distributions of viscous dissipation spectra (a1) $\xi^{x}_{kt}$
and $\xi^{x}_{\theta\theta}$ and interscale energy transports (b1)
$tr^{x}_{kt}$ and (b2) $tr^{x}_{\theta\theta}$, presented in the same manner
as in Fig. 12. The colour scales are also the same as in the corresponding
panels of Fig. 12.

More clear differences in the transport budgets of $E^{x}_{kt}$ and
$E^{x}_{\theta\theta}$ are found by investigating their viscous dissipations
and interscale transports. Figures 13(a1) and (a2) present the distributions
of the viscous dissipation spectra of the turbulent kinetic energy
$\xi^{x}_{kt}$ and the temperature fluctuation $\xi^{x}_{\theta\theta}$,
respectively. It is shown here that $\xi^{x}_{\theta\theta}$ at small
wavelengths $\lambda_{x}$ is more significant than $\xi^{x}_{kt}$. The
difference is particularly notable in the region away from the wall $y\geq
0.2$, where $\xi^{x}_{kt}$ is obviously weak as compared to the near-wall
region while $\xi^{x}_{\theta\theta}$ indicates a certain energy dissipation
at $\lambda_{x}\approx 0.25$ ($\lambda_{x}^{+}\approx 70$) in the channel-core
region. Such differences between the viscous dissipations are relatively
distinct as compared to the production spectra $pr^{x}_{kt}$ and
$pr^{x}_{\theta\theta}$ and consistent with the observations by Antonia &
AbeAntonia and Abe (2009). However, again, this might be a surprising
observation since the effect of molecular diffusion is more significant in the
temperature field as the Prandtl number $Pr$ is smaller than 1.

The interscale transport of the turbulent kinetic energy $tr^{x}_{kt}$ and the
temperature fluctuation $tr^{x}_{\theta\theta}$ are compared in Figs. 13(b1)
and (b2). Both distributions indicate that the energy is transferred basically
from relatively large wavelengths roughly in the range $1<\lambda_{x}<10$ to
smaller wavelengths in $\lambda_{x}<1$. It is, however, also clearly seen that
$tr^{x}_{\theta\theta}$ delivers the energy to smaller wavelengths than
$tr^{x}_{kt}$; in the near-wall region the peak of the energy gain by
$tr^{x}_{kt}$ is located about $\lambda_{x}=0.54$ ($\lambda_{x}^{+}=137$)
while in the $tr^{x}_{\theta\theta}$ distribution the near-wall peak of energy
gain is located around $\lambda_{x}=0.25$ ($\lambda_{x}^{+}=63$). The
difference between the $tr^{x}_{kt}$ and $tr^{x}_{\theta\theta}$ is more
distinct in the core region of the channel, as the energy gains by
$tr^{x}_{\theta\theta}$ are still clearly observed around $\lambda_{x}\approx
0.25$ compared to that by the kinetic energy transport by $tr^{x}_{kt}$.
Hence, it can be interpreted here that the different behaviours of the viscous
dissipations $\xi^{x}_{kt}$ and $\xi^{x}_{\theta\theta}$ observed in Figs.
13(a1) and (a2) are the consequences of such different energy transports by
$tr^{x}_{kt}$ and $tr^{x}_{\theta\theta}$.

Figure 14 presents further detailed balances between the production, viscous
dissipation and interscale transport terms of the spectral transport
equations, comparing those of the $E^{x}_{kt}$ and $E^{x}_{\theta\theta}$
equations at a near-wall location $y^{+}=16$ (a) and the channel centre (b).
The values of each term are scaled by the maximum values of the corresponding
production spectra. As shown in Fig. 14, at both locations, $p^{x}_{kt}$ and
$pr^{x}_{\theta\theta}$ have their peaks at similar wavelengths,
$tr^{x}_{\theta\theta}$ delivers energy to smaller wavelengths than
$tr^{x}_{kt}$, and $\xi^{x}_{\theta\theta}$ is more significant at smaller
wavelengths than $\xi^{x}_{kt}$, as already described above.

Figure 14: Comparison between transport budgets of $E^{x}_{kt}$ and
$E^{x}_{\theta\theta}$ at (a) $y^{+}=16$ and (b) channel centre. The solid
lines represent major terms in $E^{x}_{kt}$ transport, (blue) production
$pr^{x}_{kt}$, (red) viscous dissipation $\xi^{x}_{kt}$ and (yellow)
interscale transport $tr^{x}_{tk}$, and the corresponding black solid lines
are their counterparts in $E^{x}_{\theta\theta}$ transport. The yellow non-
solid lines present different ingredients of $tr^{x}_{kt}$: ‘– –’,
$tr^{x}_{uu}$; ‘$\cdots$’, $tr^{x}_{vv}$; ’– $\cdot$ –’, $tr^{x}_{ww}$.
Pressure-strain cospectrum $\pi^{x}_{uu}$ is also shown as purple dashed
lines. The values are scaled by the maximum value of the production spectra of
each transport equation.

Among the different terms shown here, the difference between the interscale
transport terms $tr^{x}_{kt}$ and $tr^{x}_{\theta\theta}$ at the near-wall
location $y^{+}=16$ is particularly notable at small wavelengths, as shown in
Fig. 14(a). While the peak of energy gain by $tr^{x}_{kt}$ is located at
$\lambda_{x}\approx 0.5$ ($\lambda_{x}^{+}\approx 130$), the profile of
$tr^{x}_{\theta\theta}$ shows a plateau in the small wavelength range around
$\lambda_{x}\approx 0.3$, and the maximum is located at $\lambda_{x}=0.25$
($\lambda_{x}^{+}\approx 60$). Corresponding to such behaviours of the
interscale transport terms, the viscous dissipation of the temperature
fluctuation $\xi^{x}_{\theta\theta}$ shows larger contributions at small
wavelengths than the kinetic energy dissipation $\xi^{x}_{kt}$.

At the channel centre, on the other hand, the difference between the
interscale and viscous dissipation of $E^{x}_{kt}$ and $E^{x}_{\theta\theta}$
are not as distinct as in the near-wall location, as shown in Fig. 14(b).
However, the same tendency is still observed that the energy is transferred to
and dissipated at relatively small wavelengths in the $E^{x}_{\theta\theta}$
transport as compared to the $E^{x}_{kt}$ transport.

In each panel of Fig. 14, the ingredients of the interscale kinetic energy
transport $tr^{x}_{kt}$, i.e. the transport of each velocity fluctuation
components $tr^{x}_{uu}$, $tr^{x}_{vv}$ and $tr^{x}_{ww}$, are also presented
for comparison. It is shown in the panel (a) that at the near-wall location
$y^{+}=16$ the interscale kinetic energy transport $tr^{x}_{kt}$ is basically
dominated by the streamwise turbulent energy transport $tr^{x}_{uu}$, the
transport of the wall-normal turbulent energy $tr^{x}_{vv}$ is negligibly
small, and the transport of the spanwise turbulent energy $tr^{x}_{ww}$ is
relatively weak compared to $tr^{x}_{uu}$ but indicates both forward and
inverse interscale energy transfers. The pressure-strain energy redistribution
spectrum $\pi^{x}_{uu}$ is also presented here, and one can see that the
profiles of $\pi^{x}_{uu}$ and $tr^{x}_{ww}$ have their negative peaks at
roughly same wavelengths, which also correspond to the peak location of the
energy gain by $tr^{x}_{uu}$. This indicates that the turbulent energy
transferred from large $\lambda_{x}$ to the middle $\lambda_{x}$ range by
$tr^{x}_{uu}$ is redistributed to $E^{x}_{ww}$ by $\pi^{x}_{uu}$, and then it
is further transferred to both smaller and larger scales by $tr^{x}_{ww}$.

In our previous studyKawata and Tsukahara (2021), the role of such interscale
transport of the spanwise turbulent energy $\left\langle w^{2}\right\rangle$
towards both larger and smaller wavelengths in the near-wall region was
closely investigated. The energy transferred to smaller wavelengths by
$tr^{x}_{ww}$ is dissipated by $\xi^{x}_{ww}$ in addition to the main energy
dissipation by $\xi^{x}_{uu}$. On the other hand, the energy “transferred
back” to larger wavelengths by $tr^{x}_{ww}$ is further transferred to the
wall-normal turbulent energy $\left\langle v^{2}\right\rangle$ by the
pressure-strain energy redistribution, and eventually leads to turbulent
energy production at the large wavelengths through the Reynolds shear stress
production. We suggested that such near-wall energy transports including
reversed energy cascade by $tr^{x}_{ww}$ may correspond to the regeneration
process of streamwise vortices in the self-sustaining cycle of the near-wall
coherent structures, which is in agreement with the suggestion by HambaHamba
(2019).

At the channel centre, on the other hand, both $tr^{x}_{vv}$ and $tr^{x}_{ww}$
present contributions comparable to $tr^{x}_{uu}$ at small wavelengths around
$\lambda_{x}\approx 0.3$, indicating forward energy cascades, as shown in Fig.
14(b) (note here that the profiles of $tr^{x}_{vv}$ and $tr^{x}_{ww}$ shown in
the panel are almost on top of each other). One can also see here that the
energy gain by $tr^{x}_{uu}$ at middle wavelengths around $\lambda_{x}\approx
1$ is almost balanced by the energy sink by $\pi^{x}_{uu}$, and $tr^{x}_{vv}$
and $tr^{x}_{ww}$ are shown to transport energy from this $\lambda_{x}$ range
to smaller wavelengths. Due to such contributions by $tr^{x}_{vv}$ and
$tr^{x}_{ww}$, the contribution by the overall kinetic energy transport
$tr^{x}_{kt}$ is comparable to the temperature-fluctuation transport
$tr^{x}_{\theta\theta}$ at small wavelengths. This is in contrast to the
transport balance in the near-wall region, where as shown in Fig. 14(a) the
energy transports by $tr^{x}_{vv}$ and $tr^{x}_{ww}$ are substantially small
compared to $tr^{x}_{uu}$, and therefore the energy transport by $tr^{x}_{kt}$
is clearly inferior to $tr^{x}_{\theta\theta}$ at small wavelengths.

As described so far, the difference between the spectral transports of
turbulent kinetic energy and temperature fluctuation is particularly found in
the behaviours of the interscale energy transport terms and, therefore, we
next look into further details of $tr^{x}_{kt}$ and $tr^{x}_{\theta\theta}$.
According to Eqs. (II.2.1) and (23), the interscale energy fluxes
$Tr^{x}_{kt}$ and $Tr^{x}_{\theta\theta}$ consist of six terms representing
the effects of different velocity or temperature gradients:

$\displaystyle Tr^{x}_{kt}=\underbrace{-\left\langle
u_{i}^{S}u^{S}\frac{\partial u_{i}^{L}}{\partial
x}\right\rangle}_{Tr^{x,(1)}_{kt}}\underbrace{-\left\langle
u_{i}^{S}v^{S}\frac{\partial u_{i}^{L}}{\partial
y}\right\rangle}_{Tr^{x,(2)}_{kt}}\underbrace{-\left\langle
u_{i}^{S}w^{S}\frac{\partial u_{i}^{L}}{\partial
z}\right\rangle}_{Tr^{x,(3)}_{kt}}$ $\displaystyle\underbrace{+\left\langle
u_{i}^{L}u^{L}\frac{\partial u_{i}^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4)}_{kt}}\underbrace{+\left\langle
u_{i}^{L}v^{L}\frac{\partial u_{i}^{S}}{\partial
y}\right\rangle}_{Tr^{x,(5)}_{kt}}\underbrace{+\left\langle
u_{i}^{L}w^{L}\frac{\partial u_{i}^{S}}{\partial
z}\right\rangle}_{Tr^{x,(6)}_{kt}},$ (30a) $\displaystyle
Tr^{x}_{\theta\theta}=\underbrace{-2\left\langle\theta^{S}u^{S}\frac{\partial\theta^{L}}{\partial
x}\right\rangle}_{Tr^{x,(1)}_{\theta\theta}}\underbrace{-2\left\langle\theta^{S}v^{S}\frac{\partial\theta^{L}}{\partial
y}\right\rangle}_{Tr^{x,(2)}_{\theta\theta}}\underbrace{-2\left\langle\theta^{S}w^{S}\frac{\partial\theta^{L}}{\partial
z}\right\rangle}_{Tr^{x,(3)}_{\theta\theta}}$
$\displaystyle\underbrace{+2\left\langle\theta^{L}u^{L}\frac{\partial\theta^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4)}_{\theta\theta}}\underbrace{+2\left\langle\theta^{L}v^{L}\frac{\partial\theta^{S}}{\partial
y}\right\rangle}_{Tr^{x,(5)}_{\theta\theta}}\underbrace{+2\left\langle\theta^{L}w^{L}\frac{\partial\theta^{S}}{\partial
z}\right\rangle}_{Tr^{x,(6)}_{\theta\theta}}.$ (30b)

Hence, the corresponding interscale transport terms in the equations of
$E^{x}_{uu}$ and $E^{x}_{\theta\theta}$ are also decomposed accordingly:

$\displaystyle tr^{x}_{kt}=\sum^{6}_{i=1}tr^{x,(i)}_{kt},\quad
tr^{x}_{\theta\theta}=\sum^{6}_{i=1}tr^{x,(i)}_{\theta\theta},$ (31)

where $tr^{x,(i)}_{kt}$ and $tr^{x,(i)}_{\theta\theta}$ are the
$k_{x}$-derivatives of the corresponding terms in Eqs. (30).

Figure 15: Comparisons of different terms in interscale energy transport terms
(a) $tr^{x}_{uu}$ and (b) $tr^{x}_{\theta\theta}$ at near-wall location
$y^{+}=16$. The values are scaled by the peak values of the corresponding
production spectra. The colours of the plots in the panel (b) presents the
same terms as in the panel (a).

The various terms in the interscale transports $tr^{x}_{kt}$ and
$tr^{x}_{\theta\theta}$ are compared for a near-wall location $y^{+}=16$ in
Fig. 15. As shown in the panel (a), among various temrs in the turbulent
kinetic energy transport the third term $tr^{x,(3)}_{kt}$ is shown to play
particularly significant role at relatively large wavelengths, accounting for
most of energy transfers from the largest wavelengths in the range
$\lambda_{x}\approx 10$ ($\lambda_{x}^{+}>1000$) to the middle wavelengths
around $\lambda_{x}\approx 1$ ($\lambda_{x}^{+}\approx 200$). This term
$tr^{x,(3)}_{kt}$ is associated with the spanwise gradient of the large-scale
fluctuating velocity $\partial u_{i}^{L}/\partial z$, as can be seen in Eq.
(30a). Therefore, the significant contribution of this term may represent the
streak instabilities consisting of the self-sustaining cycle of the near-wall
structures, as discussed in some earlier studiesKawata and Tsukahara (2021).
This may also be consistent with the observations by Wang et al.Wang, Pan, and
Wang (2020) of forward scattering events associated with large-scale motions.
The other terms of $tr^{x}_{kt}$ are shown to have relatively small
contributions compared to the third term $tr^{x,(3)}_{kt}$, indicating forward
energy transfers at certainly smaller wavelengths. Among them, the fourth term
$tr^{x,(4)}_{kt}$ provides a relatively distinct contributions, accounting for
the energy supplies to smallest wavelengths $\lambda_{x}\approx 0.22$
($\lambda_{x}^{+}\approx 50$). As shown by Eq. (30a), the fourth term
$tr^{x,(4)}_{kt}$ is associated with the streamwise gradients of small-scale
velocities $\partial u_{i}^{S}/\partial x$.

Similar trends are also found in the behaviours of decomposed interscale
transports of temperature fluctuation $tr^{x,(i)}_{\theta\theta}$. As
presented in Fig. 15(b), the third term $tr^{x,(3)}_{\theta\theta}$ dominates
the forward energy transfers from the largest wavelengths $\lambda_{x}>10$,
and the other terms show relatively small contributions, indicating forward
energy transfers towards smaller wavelengths. The distinct differences in the
behaviours of $tr^{x,(i)}_{kt}$ and $tr^{x,(i)}_{\theta\theta}$ is that the
fourth term of the temperature fluctuation transport
$tr^{x,(4)}_{\theta\theta}$ plays a significant role compared to the
counterpart of the kinetic energy transport $tr^{x,(4)}_{kt}$. As shown in
Fig. 15(b), the magnitudes of the positive and negative peaks of
$tr^{x,(4)}_{\theta\theta}$ are comparable to those of the third term
$tr^{x,(3)}_{\theta\theta}$, which is a clearly larger contribution than that
by $tr^{x,(4)}_{kt}$. Due to the significant peak of
$tr^{x,(4)}_{\theta\theta}$ in the smallest wavelength range the top of the
positive peak of the overall $tr^{x}_{\theta\theta}$ profile is somewhat
plateau-like and the maximum location is at $\lambda_{x}=0.25$
($\lambda_{x}^{+}\approx 60$). Therefore, the difference between the overall
interscale transports $tr^{x}_{kt}$ and $tr^{x}_{\theta\theta}$ results from
such significant contribution by $tr^{x,(4)}_{\theta\theta}$ in the
temperature fluctuation transport.

At the channel centre, the difference between the behaviours of the kinetic
energy transport $tr^{x}_{kt}$ and the temperature-fluctuation transport
$tr^{x}_{\theta\theta}$ is not as distinct as in the near-wall region as
already shown in Fig. 14(b), but their ingredients present certainly different
behaviours. As show in Fig. 16(a), in the kinetic energy transport the second
and third terms play the role of transferring energy from the largest
wavelengths to the middle $\lambda_{x}$ range, the fifth and sixth terms
further transfers the energy to further smaller wavelengths, and the fourth
term takes the role, again, of the energy transfer in the smallest wavelength
range. In the temperature-fluctuation transfer $tr^{x}_{\theta\theta}$, on the
other hand, while the second and third terms are shown to dominate forward
energy transfers in the largest $\lambda_{x}$ range similarly to the kinetic
energy transport $tr^{x}_{kt}$, the contributions by the fifth and sixth terms
are not significant and the fourth term is shown to take the role of
transferring the energy supplied by $tr^{x,(2)}_{\theta\theta}$ and
$tr^{x,(3)}_{\theta\theta}$ in the middle $\lambda_{x}$ range further towards
smallest wavelengths, as shown in Fig. 16(b). Such energy transfers in the
small wavelength range is in contrast to the kinetic energy transport, in
which the energy supply to the smallest wavelengths is accounted for by
composite contributions by $tr^{x,(4)}_{kt}$, $tr^{x,(5)}_{kt}$, and
$tr^{x,(6)}_{kt}$.

Figure 16: Comparisons of different terms in (a) $tr^{x}_{kt}$ and (b)
$tr^{x}_{\theta\theta}$ at the channel centre, presented in the same manner as
in Fig. 15.

It might also be worth pointing out here that each term in the kinetic energy
interscale transport is decomposed into the transports of $\left\langle
u^{2}\right\rangle$, $\left\langle v^{2}\right\rangle$, and $\left\langle
w^{2}\right\rangle$ as

$\displaystyle
tr^{x,(i)}_{kt}=tr^{x,(i)}_{uu}+tr^{x,(i)}_{vv}+tr^{x,(i)}_{ww},$ (32)

and while $tr^{x,(i)}_{uu}$ is dominant in all $tr^{x,(i)}_{kt}$ terms at the
near-wall location $y^{+}=16$, the contributions by $tr^{x,(i)}_{uu}$,
$tr^{x,(i)}_{vv}$, and $tr^{x,(i)}_{ww}$ are comparable at the channel centre
in $tr^{x,(4)}_{kt}$, $tr^{x,(5)}_{kt}$, and $tr^{x,(6)}_{kt}$, which are
responsible to the forward energy transfers at relatively small wavelengths.

###  IV.2 Spectral Analysis on Wall Shear Stress and Wall Heat Flux

As observed in Fig. 7, the integrated streamwise spectrum of the turbulent
heat transfer $\mathcal{E}^{x}_{-v\theta}$ was shown to have its peak at
relatively small $\lambda_{x}$ as compared to the integrated Reynolds shear
stress cospectrum $\mathcal{E}^{x}_{-uv}$, indicating that the streamwise
length scales at which turbulent heat transfer mainly occurs are relatively
smaller than those of turbulent momentum transfer. Now, we address this point
by investigating the transport equations of these integrated spectra
$\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$, similarly to the
analysis in the previous section. Integrating the transport equations of
$E^{x}_{-uv}$ across the channel one obtains the transport equations of
$\mathcal{E}^{x}_{-uv}$ as

$\displaystyle\frac{{\rm D}\mathcal{E}^{x}_{-uv}}{{\rm D}t}$
$\displaystyle=\mathcal{P}^{x}_{-uv}-\varXi^{x}_{-uv}+\mathcal{W}^{x}_{-uv}+\mathcal{T}^{x}_{-uv},$
(33a) where $\mathcal{P}^{x}_{-uv}$, $\varXi^{x}_{-uv}$,
$\mathcal{W}^{x}_{-uv}$, and $\mathcal{T}^{x}_{-uv}$ are the production,
viscous dissipation, pressure work, and the interscale transport terms defined
as $\displaystyle\mathcal{P}^{x}_{-uv}=\int^{1}_{0}pr^{x}_{-uv}\mathrm{d}y,$
$\displaystyle\quad\varXi^{x}_{-uv}=\int^{1}_{0}\xi^{x}_{-uv}\mathrm{d}y,$
(33b)
$\displaystyle\mathcal{W}^{x}_{-uv}=\int^{1}_{0}\phi^{x}_{-uv}\mathrm{d}y,$
$\displaystyle\quad\mathcal{T}^{x}_{-uv}=\int^{1}_{0}tr^{x}_{-uv}\mathrm{d}y.$
(33c)

Note here that the terms corresponding to the viscous diffusion
$d^{\nu,x}_{-uv}$ and the turbulent transport $d^{t,x}_{-uv}$ do not appear in
the above equations as their integrations across the channel are zero at any
wavelength. The transport equation of $\mathcal{E}^{x}_{-v\theta}$ is also
obtained by integrating the $E^{x}_{-v\theta}$ transport equation as

$\displaystyle\frac{{\rm D}\mathcal{E}^{x}_{-v\theta}}{{\rm
D}T}=\mathcal{P}^{x}_{-v\theta}-\varXi^{x}_{-v\theta}+\mathcal{W}^{x}_{-v\theta}+\mathcal{T}^{x}_{-v\theta},$
(34)

and the terms on the right-hand side are defined as the integration of the
corresponding terms in Eq. (24) similarly in Eq. (33b,c).

Figure 17: Comparison between the transport budgets of $\mathcal{E}^{x}_{-uv}$
and $\mathcal{E}^{x}_{-v\theta}$: (blue) productions, (red) viscous
dissipations, (yellow) interscale transports, and (purple) pressure works;
solid lines, $\mathcal{E}^{x}_{-uv}$ transport; dashed lines,
$\mathcal{E}^{x}_{-v\theta}$ transport. The values of the terms in the
$\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$ transport equations
are scaled by $u_{\tau}^{\ast 3}$ and $u_{\tau}^{\ast 2}T_{\tau}^{\ast}$,
respectively.

The budgets of the transport equations of $\mathcal{E}^{x}_{-uv}$ and
$\mathcal{E}^{x}_{-v\theta}$ are compared in Fig. 17. As shown here, the
production spectra $\mathcal{P}^{x}_{-uv}$ and $\mathcal{P}^{x}_{-v\theta}$
show similar profiles with their peak locations at same wavelength
$\lambda_{x}=0.8$ (corresponding to $\lambda_{x}^{+}\approx 200$). It should
be noted here that the production spectra $pr^{x}_{-uv}$ and
$pr^{x}_{-v\theta}$ both depend on the wall-normal turbulent energy spectrum
$E^{x}_{vv}$:

$\displaystyle pr^{x}_{-uv}=E^{x}_{vv}\frac{\mathrm{d}U}{\mathrm{d}y},$ (35)
$\displaystyle
pr^{x}_{-v\theta}=E^{x}_{vv}\frac{\mathrm{d}\varTheta}{\mathrm{d}y}.$ (36)

Hence, the scales at which the productions of $\mathcal{E}^{x}_{-uv}$ and
$\mathcal{E}^{x}_{-v\theta}$ mainly take place are both determined by the
$E^{x}_{vv}$ distribution.

As shown in Fig. 17, the budgets of the transport equations of
$\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$ are basically
similar. The productions are mainly balanced by the pressure work spectra
$\mathcal{W}_{-uv}$ and $\mathcal{W}_{-v\theta}$, and the profiles of
$\mathcal{W}^{x}_{-uv}$ and $\mathcal{W}^{x}_{-v\theta}$ show their peak
magnitude at the same wavelength $\lambda_{x}\approx 0.6$
($\lambda_{x}^{+}\approx 150$) similarly to those of the productions. The
viscous dissipation spectra $\varXi^{x}_{-uv}$ and $\varXi^{x}_{-v\theta}$
present only minor contributions.

The difference between the budgets of the $\mathcal{E}^{x}_{-uv}$ and
$\mathcal{E}^{x}_{-v\theta}$ equations are, again, observed in the behaviours
of the interscale transport terms $\mathcal{T}^{x}_{-uv}$ and
$\mathcal{T}^{x}_{-v\theta}$. As can be seen in Fig. 17, both
$\mathcal{T}^{x}_{-uv}$ and $\mathcal{T}^{x}_{-v\theta}$ transfer energies
from relatively large wavelengths $\lambda_{x}>2$ ($\lambda_{x}^{+}>500$) to
smaller wavelengths around $\lambda_{x}\approx 0.3$–0.5, but
$\mathcal{T}^{x}_{-v\theta}$ is shown to transfer energy to smaller
wavelengths than $\mathcal{T}^{x}_{-uv}$. Such behaviours of the interscale
transports are, therefore, attributable to the difference between the profiles
of $\mathcal{E}^{x}_{-uv}$ and $\mathcal{E}^{x}_{-v\theta}$ observed in Fig.
7(a).

Similarly to Eqs. (30), the interscale fluxes $Tr^{x}_{-uv}$ and
$Tr^{x}_{-v\theta}$ are decomposed into six different terms representing the
effects of the large- or small-scale velocity gradients in different
directions as

$\displaystyle Tr^{x}_{-uv}$ $\displaystyle=\underbrace{\left\langle
u^{s}u^{s}\frac{\partial v^{L}}{\partial x}\right\rangle+\left\langle
v^{s}u^{s}\frac{\partial u^{L}}{\partial
x}\right\rangle}_{Tr^{x,(1)}_{-uv}}\>\>\>\underbrace{+\left\langle
u^{s}v^{s}\frac{\partial v^{L}}{\partial y}\right\rangle+\left\langle
v^{s}v^{s}\frac{\partial u^{L}}{\partial
y}\right\rangle}_{Tr^{x,(2)}_{-uv}}\>\>\>\underbrace{+\left\langle
u^{s}w^{s}\frac{\partial v^{L}}{\partial z}\right\rangle+\left\langle
v^{s}w^{s}\frac{\partial u^{L}}{\partial z}\right\rangle}_{Tr^{x,(3)}_{-uv}}$
$\displaystyle\underbrace{-\left\langle u^{L}u^{L}\frac{\partial
v^{S}}{\partial x}\right\rangle-\left\langle v^{L}u^{L}\frac{\partial
u^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4)}_{-uv}}\>\>\>\underbrace{-\left\langle
u^{L}v^{L}\frac{\partial v^{S}}{\partial y}\right\rangle-\left\langle
v^{L}v^{L}\frac{\partial u^{S}}{\partial
y}\right\rangle}_{Tr^{x,(5)}_{-uv}}\>\>\>\underbrace{-\left\langle
u^{L}w^{L}\frac{\partial v^{S}}{\partial z}\right\rangle-\left\langle
v^{L}w^{L}\frac{\partial u^{S}}{\partial z}\right\rangle}_{Tr^{x,(6)}_{-uv}},$
(37a) $\displaystyle Tr^{x}_{-v\theta}$
$\displaystyle=\underbrace{\left\langle\theta^{s}u^{s}\frac{\partial
v^{L}}{\partial x}\right\rangle+\left\langle
v^{s}u^{s}\frac{\partial\theta^{L}}{\partial
x}\right\rangle}_{Tr^{x,(1)}_{-v\theta}}\>\>\>\underbrace{+\left\langle\theta^{s}v^{s}\frac{\partial
v^{L}}{\partial y}\right\rangle+\left\langle
v^{s}v^{s}\frac{\partial\theta^{L}}{\partial
y}\right\rangle}_{Tr^{x,(2)}_{-v\theta}}\>\>\>\underbrace{+\left\langle\theta^{s}w^{s}\frac{\partial
v^{L}}{\partial z}\right\rangle+\left\langle
v^{s}w^{s}\frac{\partial\theta^{L}}{\partial
z}\right\rangle}_{Tr^{x,(3)}_{-v\theta}}\>\>\>$
$\displaystyle\underbrace{-\left\langle\theta^{L}u^{L}\frac{\partial
v^{S}}{\partial x}\right\rangle-\left\langle
v^{L}u^{L}\frac{\partial\theta^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4)}_{-v\theta}}\>\>\>\underbrace{-\left\langle\theta^{L}v^{L}\frac{\partial
v^{S}}{\partial y}\right\rangle-\left\langle
v^{L}v^{L}\frac{\partial\theta^{S}}{\partial
y}\right\rangle}_{Tr^{x,(5)}_{-v\theta}}\>\>\>\underbrace{-\left\langle\theta^{L}w^{L}\frac{\partial
v^{S}}{\partial z}\right\rangle-\left\langle
v^{L}w^{L}\frac{\partial\theta^{S}}{\partial
z}\right\rangle}_{Tr^{x,(6)}_{-v\theta}}.$ (37b)

Hence, the integrated interscale transport terms $\mathcal{T}^{x}_{-uv}$ and
$\mathcal{T}^{x}_{-v\theta}$ are also decomposed into the six different
corresponding terms $\mathcal{T}^{x,(i)}_{-uv}$ and
$\mathcal{T}^{x,(i)}_{-v\theta}$ defined as

$\displaystyle\mathcal{T}^{x,(i)}_{-uv}=\int^{1}_{0}tr^{x,(i)}_{-uv}\mathrm{d}y,\quad\text{where}\quad
tr^{x,(i)}_{-uv}=-\frac{\partial Tr^{x,(i)}_{-uv}}{\partial k_{x}},$ (38a)
$\displaystyle\mathcal{T}^{x,(i)}_{-v\theta}=\int^{1}_{0}tr^{x,(i)}_{-v\theta}\mathrm{d}y,\quad\text{where}\quad
tr^{x,(i)}_{-v\theta}=-\frac{\partial Tr^{x,(i)}_{-v\theta}}{\partial k_{x}}.$
(38b)

Their contributions are compared in Fig. 18. As one can see in the panel (a),
the terms playing major roles in the Reynolds shear stress transport are the
second, third, and sixth terms, which all present forward interscale energy
transfers in different wavelength ranges: the third term
$\mathcal{T}^{x,(3)}_{-uv}$ accounts for the energy transfers in the largest
wavelength range ($\lambda\approx 0.8$–0.9), similarly to the turbulent energy
transport, and $\mathcal{T}^{x,(2)}_{-uv}$ and $\mathcal{T}^{x,(6)}_{-uv}$
represent those towards the middle and small ranges ($\lambda\approx 0.5$–0.6
and 0.3–0.4), respectively.

As for the turbulent heat flux transfer $\mathcal{T}^{x}_{-v\theta}$, the
third and the sixth terms, $\mathcal{T}^{x,(3)}_{-v\theta}$ and
$\mathcal{T}^{x,(6)}_{-v\theta}$, present major contributions similarly to
their counterparts in the Reynolds shear stress transport, but the differences
between $\mathcal{T}^{x}_{-uv}$ and $\mathcal{T}^{x}_{-v\theta}$ are indicated
by, again, the fourth term $\mathcal{T}^{x,(4)}_{-v\theta}$. As can be seen in
Fig. 18, the fourth term in the interscale transport of the Reynolds shear
stress $\mathcal{T}^{x,(4)}_{-uv}$ shows only minor contribution but the
counterpart in interscale transport of the turbulent heat transfer
$\mathcal{T}^{x,(4)}_{-v\theta}$ dominates the transports in the smallest
$\lambda_{x}$ range, similarly to the corresponding term in the temperature-
fluctuation interscale transport $tr^{x}_{\theta\theta}$.

Figure 18 also shows that the second terms $\mathcal{T}^{x,(2)}_{-uv}$ and
$\mathcal{T}^{x,(2)}_{-v\theta}$ behave differently. While
$\mathcal{T}^{x,(2)}_{-uv}$ gives the most significant energy transfer among
the ingredients of the Reynolds shear stress transport
$\mathcal{T}^{x}_{-uv}$, the contribution by $\mathcal{T}^{x,(2)}_{-v\theta}$
in the interscale energy transport is relatively small compared to other major
terms, such as $\mathcal{T}^{x,(3)}_{-v\theta}$ and
$\mathcal{T}^{x,(6)}_{-v\theta}$. Due to such differences in the contributions
by the second and fourth terms, the profile of the overall interscale transfer
$\mathcal{T}^{x}_{-v\theta}$ presents the peak at somewhat smaller wavelengths
than the Reynolds shear stress transport $\mathcal{T}^{x}_{-uv}$.

Figure 18: Comparisons between different terms in the integrated interscale
transport term (a) $\mathcal{T}^{x}_{-uv}$ and (b)
$\mathcal{T}^{x}_{-v\theta}$. The values are scaled by $u_{\tau}^{\ast 3}$ and
$u_{\tau}^{\ast 2}T_{\tau}^{\ast 2}$, respectively, and the line colours in
the panel (b) represent the same terms as in the panel (a).

Through the above discussions, the interscale transport of the turbulent heat
flux $\mathcal{T}^{x}_{-v\theta}$ has been shown to transfer more spectral
energy to smaller streamwise wavelengths than the interscale Reynolds shear
stress transport $\mathcal{T}^{x}_{-uv}$, and the fourth term has been found
to play a particularly important role in the smallest $\lambda_{x}$ range
similarly to the temperature-fluctuation transport. The fourth terms of the
interscale transport $\mathcal{T}^{x,(4)}_{-uv}$ and
$\mathcal{T}^{x,(4)}_{-v\theta}$ are, as seen in Eqs. (37a) and (37b),
comprised of two terms representing the $x$-gradients of small-scale different
velocity components or temperature:

$\displaystyle Tr^{x,(4)}_{-uv}=\underbrace{-\left\langle
u^{L}u^{L}\frac{\partial v^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4\text{-I})}_{-uv}}\;\underbrace{-\left\langle
v^{L}u^{L}\frac{\partial u^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4\text{-II}))}_{-uv}},$ (39a) $\displaystyle
Tr^{x,(4)}_{-v\theta}=\underbrace{-\left\langle\theta^{L}u^{L}\frac{\partial
v^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4\text{-I}))}_{-v\theta}}\;\underbrace{-\left\langle
v^{L}u^{L}\frac{\partial\theta^{S}}{\partial
x}\right\rangle}_{Tr^{x,(4\text{-II}))}_{-v\theta}}.$ (39b)

Figure 19: Comparisons between different terms in $\mathcal{T}^{x,(4)}_{-uv}$
and $\mathcal{T}^{x,(4)}_{-v\theta}$. The values are scaled by $u_{\tau}^{\tau
3}$ and $u_{\tau}^{\tau 2}T_{\tau}^{\ast}$.

All four terms are compared in Fig. 19. The most significant term is found to
be $\mathcal{T}^{x,(4\text{-II})}_{-v\theta}$, which is related to the small-
scale fluctuating temperature gradient $\partial\theta^{S}/\partial x$. This
term is clearly responsible to the notable contribution by the fourth term
$\mathcal{T}^{x,(4)}_{-v\theta}$ at the small wavelengths. This trend is
similar to the interscale transport of the temperature fluctuation
$tr^{x}_{\theta\theta}$, in that the significant energy transport to the
smallest $\lambda_{x}$ range by $tr^{x}_{\theta\theta}$ is also mainly due to
the term associated with $\partial\theta^{S}/\partial x$, as shown in the
previous section.

Thus, through this section, it has been revealed that the interscale energy
transfers of temperature fluctuation at smallest $\lambda_{x}$ range is
dominated by the term related with the streamwise temperature gradient, and
the energy transport by this term has been found notably significant as
compared to the corresponding term of the turbulent kinetic energy transport.
Such significant interscale transport in the temperature-fluctuation spectrum
transport is mainly responsible to the temperature fluctuation at smaller
$\lambda_{x}$ than the velocity fluctuation and the certain scale-by-scale
dissimilarity between the turbulent momentum and heat transfers.

##  V Conclusion

In the present study, a turbulent plane Couette flow with passive-scalar heat
transfer has been simulated in order to investigate the interscale and spatial
transports of the temperature-related statistics in comparisons with the
corresponding transports of the turbulent kinetic energy and the Reynolds
shear stress. Due to the similar boundary conditions in the velocity and
temperature fields, the profiles of the mean streamwise velocity and the mean
temperature are similar to each other, and the productions of the turbulent
energy and the temperature fluctuation, therefore, also have similar profiles.
In such a configuration, the distributions of the spectra and their transport
budgets have been investigated in details.

It has been found that the temperature fluctuation occurs at somewhat smaller
streamwise length scales than the velocity fluctuations, although though the
Prandtl number is smaller than 1 and therefore the effect of molecular
diffusion is more significant in the temperature field. The turbulent heat
flux spectrum is also found to be distributed in smaller streamwise wavelength
range than the Reynolds shear stress cospectrum, indicating a certain scale-
by-scale dissimilarity between the turbulent momentum and heat transfers. The
transport budget analysis of the temperature-related spectra reveals that the
interscale energy transport of the temperature fluctuation is notably
significant compared to that of the turbulent kinetic energy at smallest
streamwise wavelengths. Among various terms comprising the interscale energy
transport, the term representing the effect by the streamwise temperature
gradient at small scales is found to dominate the transport of the temperature
fluctuation to the smallest wavelength range, while the counterpart in the
turbulent kinetic energy transport plays only a minor role in the energy
transport budget. Such difference between the interscale transports of the
turbulent kinetic energy and the temperature fluctuation is attributable to
the different distributions of their spectra at small scales. The same
tendency is also found in the interscale transports of the Reynolds shear
stress and the turbulent heat flux, which results in the spectral
dissimilarity between the turbulent heat and momentum transfers.

###### Acknowledgements.

This work was supported by the Japan Society for the Promotion of Science
(JSPS) through JSPS KAKENHI, Grant No. JP20K14654. The numerical simulations
in the present study were performed by SX-ACE supercomputers at the Cybermedia
Center of Osaka University.

## Author Declaration

### Conflict of Interest

The authors have no conflicts to disclose.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Smits, McKeon, and Marusic (2011) A. Smits, B. McKeon, and I. Marusic, “High–Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
  * Mathis, Hutchins, and Marusic (2009) R. Mathis, N. Hutchins, and I. Marusic, “Large-scale amplitude modulation of the small-scale structres in turbulent boundary layers,” J. Fluid Mech. 628, 311–337 (2009).
  * Bernardini and Pirozzoli (2011) M. Bernardini and S. Pirozzoli, “Inner/outer layer interactions in turbulent boundary layers: A refined measure for the large-scale amplitude modulation mechanism,” Phys. Fluids 23, 061701 (2011).
  * Pirozzoli, Bernardini, and Orlandi (2011) S. Pirozzoli, M. Bernardini, and P. Orlandi, “Large-scale motions and inner/outer layer interactions in turbulent Couette–Poiseuille flows,” J. Fluid Mech. 680, 534–563 (2011).
  * Dogan  _et al._ (2019) E. Dogan, R. Örlü, D. Gatti, R. Vinuesa, and P. Schlatter, “Quantification of amplitude modulation in wall-bounded turbulence,” Fluid Dyn. Research 51, 011408 (2019).
  * Cimarelli  _et al._ (2016) A. Cimarelli, E. De Angelis, J. Jiménez, and C. M. Casciola, “Cascades and wall-normal fluxes in turbulent channel flows,” J. Fluid Mech. 796, 417–436 (2016).
  * Mizuno (2016) Y. Mizuno, “Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers,” J. Fluid Mech. 805, 171–187 (2016).
  * Mamatsashvili  _et al._ (2016) G. Mamatsashvili, G. Khujadze, G. Chagelishvili, S. Dong, J. Jiménez, and H. Foysi, “Dynamics of homogeneous shear turbulence: Akey role of the nonlinear transverse cascade in the bypass concept,” Phys. Rev. E 94, 023111 (2016).
  * Cho, Hwang, and Choi (2018) M. Cho, Y. Hwang, and H. Choi, “Scale interactions and spectral energy transfer in turbulent channel flow,” J. Fluid Mech. 854, 474–504 (2018).
  * Kawata and Alfredsson (2018) T. Kawata and P. H. Alfredsson, “Inverse interscale transport of the Reynolds shear stress in plane Couette turbulence,” Phys. Rev. Lett. 120, 244501 (2018).
  * Bauer, von Kameke, and Wagner (2019) C. Bauer, A. von Kameke, and C. Wagner, “Kinetic energy budget of the largest scales in turbulent pipe flow,” Phys. Fluids 4, 064607 (2019).
  * Hamba (2019) F. Hamba, “Inverse energy cascade and vortical structure in the near-wall region of turbulent channel flow,” Phys. Rev. Fluids 4, 114609 (2019).
  * Lee and Moser (2019) M. Lee and R. D. Moser, “Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number,” J. Fluid Mech. 860, 886–938 (2019).
  * Motoori and Goto (2019) Y. Motoori and S. Goto, “Generation mechanism of a hierarchy of vortices in a turbulent boundary layer,” J. Fluid Mech. 865, pp. 1085–1109 (2019).
  * Motoori and Goto (2020) Y. Motoori and S. Goto, “Hierarchy of coherent structures in turbulent channel flow,” J. Phys.: Conf. Ser. 1522, 012004 (2020).
  * Chan, Schlatter, and Chin (2021) C. Chan, P. Schlatter, and R. Chin, “Interscale transport mechanisms in turbulent boundary layers,” J. Fluid Mech. 921, A13 (2021).
  * Chiarini  _et al._ (2021) A. Chiarini, M. Mauriello, D. Gatti, and M. Quadrio, “Ascending-descending and direct-inverse cascades of reynolds stresses in turbulent Couette flow,” J. Fluid Mech. 930, A9 (2021).
  * Doohan, Wills, and Hwang (2021) P. Doohan, A. P. Wills, and Y. Hwang, “Minimal multi-scale dynamics of near-wall turbulence,” J. Fluid Mech. 913, A8 (2021).
  * Kawata and Tsukahara (2021) T. Kawata and T. Tsukahara, “Scale interactions in turbulent plane Couette flows in minimai domains,” J. Fluid Mech. 911, A55 (2021).
  * Wang  _et al._ (2021) H. Wang, Z. Yang, T. Wu, and S. Wang, “Coherent structures associated with interscale energy transfer in turbulent channel flows,” Phys. Rev. Fluids 6, 104601 (2021).
  * Mäteling and Schröder (2022) E. Mäteling and W. Schröder, “Analysis of spatiotemporal inner-outer large-scale interactions in turbulent channel flow by multivariate empirical mode decomposition,” Phys. Rev. Fluids 7, 034603 (2022).
  * Chan  _et al._ (2022) C. Chan, R. Örlü, P. Schlatter, and R. Chin, “Large-scale and small-scale contribution to the skin friction reduction in a modified turbulent boundary layer by a large-eddy break-up device,” Phys. Rev. Fluids 7, 034601 (2022).
  * Zhou, Xu, and Jiménez (2022) Z. Zhou, C.-X. Xu, and J. Jiménez, “Interaction between near-wall streaks and large-scale motions in turbulent channel flows,” arXiv Preprint , arXiv:2203.08043 (2022).
  * Hasegawa and Kasagi (2011) Y. Hasegawa and N. Kasagi, “Dissimilar control of momentum and heat transfer in a fully developed turbulent channel flow,” J. Fluid Mech. 683, 57–93 (2011).
  * Yamamoto, Hasegawa, and Kasagi (2013) A. Yamamoto, Y. Hasegawa, and N. Kasagi, “Optimal control of dissimilar heat and momentum transfer in a fully developed turbulent channel flow,” J. Fluid Mech. 733, 189–220 (2013).
  * Higashi, Mamori, and Fukagata (2011) K. Higashi, H. Mamori, and K. Fukagata, “Simultaneous control of friction drag reduction and heat transfer augmentation by traveling wave-like blowing/suction,” Comput. Thermal Sci. 3, 521–530 (2011).
  * Kaithakkal, Kametani, and Hasegawa (2020) A. Kaithakkal, Y. Kametani, and Y. Hasegawa, “Dissimilarity between turbulent heat and momentum transfer induced by a streamwise travelling wave of wall blowing and suction,” J. Fluid Mech. 886, A29 (2020).
  * Mamori  _et al._ (2021) H. Mamori, K. Fukudome, K. Ogino, N. Fukushima, and M. Yamamoto, “Heat transfer enhancement and torque reduction by traveling wave-like blowing and suction in turbulent Taylor–Couette flow,” J. Thermal Sci. Technol. 16, 20–00256 (2021).
  * Uchino, Mamori, and Fukagata (2017) K. Uchino, H. Mamori, and K. Fukagata, “Heat transfer in fully developed turbulent channel flow with streamwise traveling wave-like wall deformation,” J. Thermal Sci. Technol. 12, 16–00624 (2017).
  * Antonia and Abe (2009) R. A. Antonia and H. Abe, “Analogy between velocity and scalar fields in a turbulent channel flow,” J. Fluid Mech. 628, 241–268 (2009).
  * Hane  _et al._ (2006) S. Hane, T. Tsukahara, K. Iwamoto, and H. Kawamura, “DNS of turbulent heat transfer in plane Couette flow,” in  _Proc. International Heat Transfer Conference 13_ , DOI: 10.1615/IHTC13.p1.200 (2006).
  * Bech  _et al._ (1995) K. H. Bech, N. Tillmark, P. H. Alfredsson, and H. I. Andersson, “An investigation of turbulent plane Couette flow at low Reynolds numbers,” J. Fluid Mech. 286, 291–325 (1995).
  * Komminaho, Lundbladh, and Johansson (1996) J. Komminaho, A. Lundbladh, and A. V. Johansson, “Very large structures in plane turbulent Couette flow,” J. Fluid Mech. 320, 259–285 (1996).
  * Kitoh, Nakabayashi, and Nishimura (2005) O. Kitoh, K. Nakabayashi, and F. Nishimura, “Experimental study on mean velocity and turbulence characteristics of plane Couette flow: low-Reynolds-number effects and large longitudinal vortical structure,” J. Fluid Mech. 539, 199–227 (2005).
  * Tsukahara, Kawamura, and Shingai (2006) T. Tsukahara, H. Kawamura, and K. Shingai, “DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region,” J. Turbulence 7, N19 (2006).
  * Kitoh and Umeki (2008) O. Kitoh and M. Umeki, “Experimental study on large-scale streak structure in the core region of turbulent plane Couette flow,” Phys. Fluids 20, 025107 (2008).
  * Avsarkisov  _et al._ (2015) V. Avsarkisov, S. Hoyas, M. Oberlack, and J. P. García-Galache, “Turbulent plane Couette flow at moderately high Reynolds number,” J. Fluid Mech. 751, R1 (2015).
  * Orlandi, Bernardini, and Pirozzoli (2015) P. Orlandi, M. Bernardini, and S. Pirozzoli, “Poiseuille and Couette flows in the transitional and fully turbulent regime,” J. Fluid Mech. 770, 424 – 441 (2015).
  * Lee and Moser (2018) M. Lee and R. D. Moser, “Extreme-scale motions in turbulent plane Couette flows,” J. Fluid Mech. 842, 128–145 (2018).
  * Kawata and Alfredsson (2019) T. Kawata and P. H. Alfredsson, “Scale interactions in turbulent rotating planar Couette flow: insight through the Reynolds stress transport,” J. Fluid Mech. 879, 255 – 295 (2019).
  * Fukagata, Iwamoto, and Kasagi (2002) K. Fukagata, K. Iwamoto, and N. Kasagi, “Contribution of reynolds stress distribution to the skin friction in wall-bounded flows stress distribution to the skin friction in wall-bounded flows,” Phys. Fluids 14, L73–L76 (2002).
  * Wang, Pan, and Wang (2020) W. Wang, C. Pan, and J. Wang, “Energy transfer structures associated with large-scale motions in a turbulent boundary layer,” J. Fluid Mech. 906, A14 (2020).