# Analysis of anisotropic subgrid-scale stress
for coarse large-eddy simulation

Kazuhiro Inagaki (稲垣 和寛)  Department of Mechanical Engineering, Doshisha
University,1-3 Tatara Miyakodani, Kyotanabe 610-0394, Japan. E-mail:
<EMAIL_ADDRESS>Research and Education Center for Natural
Sciences, Keio University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan
Hiromichi Kobayashi (小林 宏充)  Department of Physics & Research and Education
Center for Natural Sciences, Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi,
Kohoku-ku, Yokohama 223-8521, Japan

###### Abstract

This study discusses the necessity of anisotropic subgrid-scale (SGS) stress
in large-eddy simulations (LESs) of turbulent shear flows using a coarse grid
resolution. We decompose the SGS stress into two parts to observe the role of
SGS stress in turbulent shear flows in addition to the energy transfer between
grid-scale (GS or resolved scale) and SGS. One is the isotropic eddy-viscosity
term, which contributes to energy transfer, and the other is the residual
anisotropic term, which is separated from the energy transfer. We investigate
the budget equation for GS Reynolds stress in turbulent channel flows
accompanied by the SGS stress decomposition. In addition, we examine the
medium and coarse filter length cases; the conventional eddy-viscosity models
can fairly predict the mean velocity profile for the medium filter case and
fails for the coarse filter case. The budget for GS turbulent kinetic energy
shows that the anisotropic SGS stress has a negligible contribution to energy
transfer. In contrast, the anisotropic stress has a large and non-dissipative
contribution to the streamwise and spanwise components of GS Reynolds stress
when the filter size is large. Even for the medium-size filter case, the
anisotropic stress contributes positively to the budget for the spanwise GS
Reynolds stress. Spectral analysis of the budget reveals that the positive
contribution is prominent at a scale consistent with the spacing of streaks in
the near-wall region. Therefore, we infer that anisotropic stress contributes
to the generation mechanism of coherent structures. Predicting the positive
contribution of the anisotropic stress to the budget is key to further
improving SGS models.

###### pacs:

Valid PACS appear here

##  I Introduction

Subgrid-scale (SGS, or subfilter-scale) models play a significant role in
accurately predicting the statistics of turbulent flows in large-eddy
simulations (LESs). For instance, eddy-viscosity models provide a
statistically accurate energy transfer rate from grid scale (GS or resolved
scale) to SGS. However, an accurate energy transfer rate is only one of the
required conditions for predicting the statistics of turbulent flows. In
general, characteristic quantities other than the energy transfer rate are
important in predicting turbulent flows; for example, the mean SGS shear
stress in turbulent shear flows. Several studies pointed out that the eddy-
viscosity models are not necessarily sufficient for predicting both the energy
transfer rate and mean SGS stress in turbulent shear flows [1, 2, 3].
Therefore, it is important to understand the role of SGS stress in addition to
energy transfer for the further development of LES.

Several studies addressed the development of non-eddy-viscosity or anisotropic
SGS models and showed that these models improve the prediction of the
statistics of turbulent flows [4, 5, 6, 7, 8, 9, 10, 11]. Specifically, these
anisotropic models provide a better prediction even in grid resolution cases
that are coarser than in conventional eddy-viscosity models. Therefore, we
infer that the difference between eddy-viscosity-based and anisotropic models
becomes prominent in coarse grid cases. Analysis using a coarse grid or large
filter scale will shed light on the role of anisotropic SGS stress in LES.
Furthermore, the necessity of improving the SGS model in a coarse grid
resolution has recently been discussed for atmospheric turbulence as the grey
zone or terra incognita problem [12].

It is worth noting that some anisotropic models prohibit backward scatter,
which is the local energy transfer from SGS to GS [4, 5, 6, 7, 8, 9]. This
indicates that additional stress, apart from energy transfer, can improve the
performance of LES. Actually, Abe [13] demonstrated that the additional stress
apart from the energy transfer essentially contributes to the generation of GS
or resolved scale Reynolds shear stress in turbulent channel flows. Based on a
similar analysis, Inagaki and Kobayashi [8] suggested that the amplification
of small-scale velocity fluctuations close to the cut-off scale due to the
anisotropic stress contributes to the improvement of the prediction of
statistics in LES. As suggested by these studies, an analysis of the budget
for the Reynolds stress will shed light on the physical role of anisotropic
SGS stress in a statistical sense.

In this study, we investigate the budget for GS Reynolds stress in turbulent
channel flows. To clarify the physical role of the anisotropy of SGS stress,
we decompose the SGS stress according to the description by Abe [13]. The
eddy-viscosity coefficient is determined by assuming that all energy transfers
between GS and SGS are governed by the eddy-viscosity term. Residual stress is
obtained by subtracting the eddy-viscosity term from the exact SGS stress.
Therefore, the residual stress is apart from the energy transfer. Because the
eddy-viscosity approximation is often referred to as the isotropic model owing
to the scalar coefficient, the residual stress represents the anisotropy of
the turbulent velocity fluctuation in SGS. Such anisotropic stress is
essential for predicting the anisotropy of the SGS dissipation tensor
(hereafter often referred to as SGS dissipation, simply), which is the key
ingredient in this study.

The anisotropy of SGS dissipation is rarely considered in the context of SGS
modeling [14]. Haering et al. [15] demonstrated that a simple tensor-
coefficient eddy-viscosity model can improve the prediction of SGS dissipation
anisotropy induced by anisotropic grids. Marstorp et al. [4] reported that LES
employing anisotropic SGS stress improves the prediction of the ratio of wall-
normal to streamwise components of SGS dissipation when compared with the
eddy-viscosity model in rotating channel flows. In conventional non-rotating
channel flow, Domaradzki et al. [16] and Härtel and Kleiser [17] showed that
in the energy transfer between GS and SGS, the streamwise component is
dominant and others are negligible in the near-wall region. As shown later,
the anisotropic property of SGS dissipation cannot be reproduced solely by the
eddy-viscosity term even though the total SGS dissipation, which is the trace
part of SGS dissipation tensor, is described adequately by the isotropic eddy
viscosity. Therefore, this study quantifies the importance of the anisotropic
SGS stress in predicting the statistics of turbulent flows in LES.

The rest of this paper is organized as follows. First, we summarize the
decomposition of SGS stress according to the description by Abe [13] and
budget equation for GS Reynolds stress in Sec. II. We also provide a physical
interpretation of SGS stress-related terms in the budget. The numerical
results for the budgets in turbulent channel flows are presented in Sec. III.
In Sec. IV, further analysis of the budgets in Fourier space is presented to
discuss the physical role of anisotropic SGS stress in the near-wall region.
In addition, we provide an a priori test of the SGS dissipation based on
anisotropic SGS stress using a few existing model expressions. Conclusions are
presented in Sec V.

##  II Budget equation for GS Reynolds stress

###  II.1 Decomposition of SGS stress

In LES of incompressible flows, the governing equations are filtered
continuity and Navier-Stokes equations:

$\displaystyle\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0,$ (1)
$\displaystyle\frac{\partial\overline{u}_{i}}{\partial
t}=-\frac{\partial}{\partial
x_{j}}(\overline{u}_{i}\overline{u}_{j}+\tau^{\mathrm{sgs}}_{ij})-\frac{\partial\overline{p}}{\partial
x_{i}}+\frac{\partial}{\partial x_{j}}(2\nu\overline{s}_{ij}),$ (2)

where $\overline{\cdot}$ represents the spatial filtering operation. Here, we
assume that the filter and partial differential operations are always
commutative. If the filter and partial differential operations are not
commutative, several additional terms appear in the filtered continuity and
Navier–Stokes equations (see e.g. Ref. [14]), which makes the investigation of
the effects of SGS on GS much more complex. To focus on the role of SGS stress
in the governing equations, we assume the commutativity of filter operation.
$\overline{u}_{i}$, $\overline{p}$, and
$\overline{s}_{ij}[=(\partial\overline{u}_{i}/\partial
x_{j}+\partial\overline{u}_{j}/\partial x_{i})/2]$ are the GS velocity,
pressure divided by the density, and strain-rate tensor, respectively. $\nu$
is the kinematic viscosity. The sole unknown variable in LES is the SGS stress
defined by
$\tau^{\mathrm{sgs}}_{ij}=\overline{u_{i}u_{j}}-\overline{u}_{i}\overline{u}_{j}$.

To numerically solve Eqs. (1) and (2), we must employ a closed model
expression for $\tau^{\mathrm{sgs}}_{ij}$. In contrast, in the analysis based
on direct numerical simulation (DNS) data, which is referred to as the a
priori test, we can directly calculate $\tau^{\mathrm{sgs}}_{ij}$ by explicit
filtering. However, in the case of an a priori test, we cannot assess the
amount of SGS stress that can be modeled in terms of the eddy-viscosity
approximation. The inconsistency between the a priori and a posteriori tests
of the SGS model increases the complexity of this issue. The correlation
between SGS stress and strain rate in the a priori test is low [18, 19, 20].
Furthermore, abundant backward scatter events [21, 22] reject the validity of
an eddy-viscosity model with a positive coefficient. Nevertheless, the purely
dissipative eddy-viscosity models perform well in several turbulent flows in
the a posteriori test of LES (see Ref. [14]).

The eddy-viscosity term is ideal to achieve numerical stability, although its
justification is still under discussion. In particular, when we employ the
scale-similarity model, the eddy-viscosity term is often employed to
compensate for insufficient dissipation [23, 24]. Furthermore, recent studies
on anisotropic SGS modeling demonstrated that the employment of the additional
stress that has no contribution to the energy transfer improves the prediction
of the statistics in LES [4, 5, 6, 7, 8, 9]. Therefore, we adopt the following
decomposition of SGS stress by assuming that the eddy-viscosity term governs
the energy transfer between GS and SGS [13, 8]:

$\displaystyle\tau^{\mathrm{sgs}}_{ij}-\frac{1}{3}\tau^{\mathrm{sgs}}_{\ell\ell}\delta_{ij}=-2\nu^{\mathrm{sgs}}\overline{s}_{ij}+\tau^{\mathrm{ani}}_{ij},$
(3)

where

$\displaystyle\nu^{\mathrm{sgs}}=-\frac{\tau^{\mathrm{sgs}}_{ij}\overline{s}_{ij}}{2\overline{s}^{2}},$
(4)
$\displaystyle\tau^{\mathrm{ani}}_{ij}=\tau^{\mathrm{sgs}}_{ij}-\frac{1}{3}\tau^{\mathrm{sgs}}_{\ell\ell}\delta_{ij}+2\nu^{\mathrm{sgs}}\overline{s}_{ij},$
(5)

and $\overline{s}^{2}=\overline{s}_{ij}\overline{s}_{ij}$. According to Abe
[13], we refer to $\tau^{\mathrm{ani}}_{ij}$ as the extra anisotropic or
simply the anisotropic term. Based on the definition of eddy viscosity given
by Eq. (4), the anisotropic term does not contribute to the energy transfer
between GS and SGS, that is, $\tau^{\mathrm{ani}}_{ij}\overline{s}_{ij}=0$.
Although this decomposition is arbitrary, this analysis allows us to verify
the physical properties of the anisotropic part of SGS stress, in addition to
the energy transfer. The eddy-viscosity defined by Eq. (4) can be negative,
which indicates the local backward scatter of kinetic energy. However, the
average energy transfer rate is almost positive because of the predominance of
forward scatter in turbulent flows. It is worth noting that this eddy-
viscosity does not necessarily predict the mean SGS shear stress in turbulent
shear flows even though it accounts for an accurate energy transfer rate [1,
2, 3]. Hence, the present analysis also provides the difference between the
exact mean SGS shear stress and that predicted only by the eddy-viscosity term
based on energy transfer.

###  II.2 Budget equation for GS Reynolds stress

Under Reynolds decomposition, $\langle\overline{q}\rangle=Q$ and
$\overline{q}^{\prime}=\overline{q}-\langle\overline{q}\rangle=\overline{q}-Q$
for $q=(u_{i},p,s_{ij})$ with the ensemble average $\langle\cdot\rangle$, the
budget equation for GS Reynolds stress
$R^{\mathrm{GS}}_{ij}(=\langle\overline{u}_{i}^{\prime}\overline{u}_{j}^{\prime}\rangle)$
in the LES reads

$\displaystyle\frac{\partial R^{\mathrm{GS}}_{ij}}{\partial
t}+\frac{\partial}{\partial
x_{\ell}}(U_{\ell}R^{\mathrm{GS}}_{ij})=P^{\mathrm{GS}}_{ij}-\varepsilon^{\mathrm{GS}}_{ij}+\Phi^{\mathrm{GS}}_{ij}+D^{\mathrm{p,GS}}_{ij}+D^{\mathrm{t,GS}}_{ij}+D^{\mathrm{v,GS}}_{ij}-\varepsilon^{\mathrm{SGS}}_{ij}+D^{\mathrm{SGS}}_{ij},$
(6)

where

$\displaystyle P^{\mathrm{GS}}_{ij}$
$\displaystyle=-R^{\mathrm{GS}}_{i\ell}\frac{\partial U_{j}}{\partial
x_{\ell}}-R^{\mathrm{GS}}_{j\ell}\frac{\partial U_{i}}{\partial x_{\ell}},$
(7a) $\displaystyle\varepsilon^{\mathrm{GS}}_{ij}$
$\displaystyle=2\nu\left<\overline{s}_{i\ell}^{\prime}\frac{\partial\overline{u}_{j}^{\prime}}{\partial
x_{\ell}}+\overline{s}_{j\ell}^{\prime}\frac{\partial\overline{u}_{i}^{\prime}}{\partial
x_{\ell}}\right>,$ (7b) $\displaystyle\Phi^{\mathrm{GS}}_{ij}$
$\displaystyle=2\left<\overline{p}^{\mathrm{total}}\overline{s}_{ij}^{\prime}\right>,$
(7c) $\displaystyle D^{\mathrm{p,GS}}_{ij}$
$\displaystyle=-\frac{\partial}{\partial
x_{\ell}}\left<\overline{p}^{\mathrm{total}}\overline{u}_{j}^{\prime}\delta_{i\ell}+\overline{p}^{\mathrm{total}}\overline{u}_{j}^{\prime}\delta_{i\ell}\right>,$
(7d) $\displaystyle D^{\mathrm{t,GS}}_{ij}$
$\displaystyle=-\frac{\partial}{\partial
x_{\ell}}\left<\overline{u}_{\ell}^{\prime}\overline{u}_{i}^{\prime}\overline{u}_{j}^{\prime}\right>,$
(7e) $\displaystyle D^{\mathrm{v,GS}}_{ij}$
$\displaystyle=2\nu\frac{\partial}{\partial
x_{\ell}}\left<\overline{s}_{i\ell}^{\prime}\overline{u}_{j}^{\prime}+\overline{s}_{j\ell}^{\prime}\overline{u}_{j}^{\prime}\right>,$
(7f) $\displaystyle\varepsilon^{\mathrm{SGS}}_{ij}$
$\displaystyle=-\left<\tau^{\mathrm{sgs}}_{i\ell}|_{\mathrm{tl}}\frac{\partial\overline{u}_{j}^{\prime}}{\partial
x_{\ell}}+\tau^{\mathrm{sgs}}_{j\ell}|_{\mathrm{tl}}\frac{\partial\overline{u}_{i}^{\prime}}{\partial
x_{\ell}}\right>,$ (7g) $\displaystyle D^{\mathrm{SGS}}_{ij}$
$\displaystyle=-\frac{\partial}{\partial
x_{\ell}}\left<\tau^{\mathrm{sgs}}_{i\ell}|_{\mathrm{tl}}\overline{u}_{j}^{\prime}+\tau^{\mathrm{sgs}}_{j\ell}|_{\mathrm{tl}}\overline{u}_{i}^{\prime}\right>,$
(7h)

$\overline{p}^{\mathrm{total}}=\overline{p}+\tau^{\mathrm{sgs}}_{mm}/3$, and
$\tau^{\mathrm{sgs}}_{ij}|_{\mathrm{tl}}=\tau^{\mathrm{sgs}}_{ij}-\tau^{\mathrm{sgs}}_{mm}\delta_{ij}/3$.
The terms expressed in Eqs. (7a)-(7h) are referred to as production,
dissipation, pressure redistribution, pressure diffusion, turbulent diffusion,
viscous diffusion, SGS dissipation, and SGS diffusion, respectively.
Contributions from SGS stress appear in the SGS dissipation
$\varepsilon^{\mathrm{SGS}}_{ij}$ and diffusion $D^{\mathrm{SGS}}_{ij}$. Note
that SGS dissipation is not necessarily positive, even for the trace part
$\varepsilon^{\mathrm{SGS}}(=\varepsilon^{\mathrm{SGS}}_{ii}/2)$, in contrast
to the molecular dissipation $\varepsilon(=\varepsilon_{ii}/2)$. Negative SGS
dissipation is observed near the wall in turbulent channel flows [16, 25, 17,
26]. In this study, we do not focus on modeling the eddy viscosity, which
predicts the spatial profile of SGS dissipation. Instead, we extract the
physical properties required for anisotropic SGS stress by the decomposition
(3) with (4). This analysis demonstrates the general property of anisotropic
SGS stress; that is, it does not rely on a specific model expression.

###  II.3 Decomposition of SGS dissipation

Using Eq. (3), we decompose the SGS dissipation
$\varepsilon^{\mathrm{SGS}}_{ij}$ into the following forms:

$\displaystyle\varepsilon^{\mathrm{SGS}}_{ij}$
$\displaystyle=\varepsilon^{\mathrm{EV}}_{ij}-\xi^{\mathrm{AR}}_{ij},$ (8)

where

$\displaystyle\varepsilon^{\mathrm{EV}}_{ij}$
$\displaystyle=2\left<\nu^{\mathrm{sgs}}\left(\overline{s}_{i\ell}\frac{\partial\overline{u}_{j}^{\prime}}{\partial
x_{\ell}}+\overline{s}_{j\ell}\frac{\partial\overline{u}_{i}^{\prime}}{\partial
x_{\ell}}\right)\right>,$ (9a) $\displaystyle\xi^{\mathrm{AR}}_{ij}$
$\displaystyle=\left<\tau^{\mathrm{ani}}_{i\ell}\frac{\partial\overline{u}_{j}^{\prime}}{\partial
x_{\ell}}+\tau^{\mathrm{ani}}_{j\ell}\frac{\partial\overline{u}_{i}^{\prime}}{\partial
x_{\ell}}\right>.$ (9b)

We refer to the terms defined in Eqs. (9a) and (9b) as eddy-viscosity
dissipation and anisotropic redistribution, respectively. The eddy-viscosity
dissipation $\varepsilon^{\mathrm{EV}}_{ij}$ (9a) has a form similar to that
of the conventional molecular dissipation $\varepsilon_{ij}$ (7b). Hence, we
expect that the eddy-viscosity dissipation plays a dissipative role similar to
the molecular one. In contrast, the anisotropic redistribution
$\xi^{\mathrm{AR}}_{ij}$ (9b) plays a different role in contrast to the eddy-
viscosity dissipation (9a) owing to the anisotropy of SGS stress. Therefore,
the anisotropy of SGS stress plays a significant role in predicting turbulent
flows when the anisotropic redistribution contributes significantly to the GS
Reynolds stress budget. Similarly, we can decompose the SGS diffusion
$D^{\mathrm{SGS}}_{ij}$ (7h) into eddy-viscosity and anisotropic parts. For
simplicity, we discuss the result of the decomposition of SGS dissipation. The
decomposition of SGS diffusion is briefly discussed in Appendix A.

The naming of anisotropic redistribution is based on the property that
$\tau^{\mathrm{ani}}_{ij}$ has no contribution to the energy transfer between
GS and SGS. In addition, we do not refer to this term as anisotropic
dissipation because we expect it to be more prominently non-dissipative than
SGS dissipation $\varepsilon^{\mathrm{SGS}}_{ij}$. Exactly speaking, however,
the anisotropic redistribution does not redistribute the energy among the
normal stress components, in contrast to the pressure redistribution
$\Phi^{\mathrm{GS}}_{ij}$ (7c). In other words, the trace of pressure
redistribution is exactly zero
$\Phi^{\mathrm{GS}}_{ii}=2\langle\overline{p}^{\mathrm{total}}\overline{s}_{ii}^{\prime}\rangle=0$,
whereas the anisotropic redistribution is not necessarily traceless; that is,
$\xi^{\mathrm{AR}}_{ii}=2\langle\tau^{\mathrm{ani}}_{ij}\overline{s}_{ij}^{\prime}\rangle\neq
0$ [8]. This non-zero trace emanates from the Reynolds decomposition; that is,

$\displaystyle\left<\tau^{\mathrm{ani}}_{ij}\overline{s}_{ij}\right>=\left<\tau^{\mathrm{ani}}_{ij}\right>S_{ij}+\xi^{\mathrm{AR}}_{ii}/2=0,$
(10)

and hence $\xi^{\mathrm{AR}}_{ii}=-\langle\tau^{\mathrm{ani}}_{ij}\rangle
S_{ij}\neq 0$. Here, we used the property
$\tau^{\mathrm{ani}}_{ij}\overline{s}_{ij}=0$, which is provided in Sec. II.1.
$\langle\tau^{\mathrm{ani}}_{ij}\rangle S_{ij}$ represents the energy transfer
rate between the mean and SGS fields due to the anisotropic stress. For
unidirectional turbulent shear flows as channel flows, it reads
$\langle\tau^{\mathrm{ani}}_{xy}\rangle\partial U_{x}/\partial y/2$. When
$\partial U_{x}/\partial y>0$ and $\langle\tau^{\mathrm{ani}}_{xy}\rangle<0$
as the conventional turbulent shear stress in shear flows,
$\xi^{\mathrm{AR}}_{ii}>0$, thus implying that the trace of anisotropic
redistribution is productive. Conversely, when $\partial U_{x}/\partial y>0$
and $\langle\tau^{\mathrm{ani}}_{xy}\rangle>0$, $\xi^{\mathrm{AR}}_{ii}<0$,
thus implying that the trace of anisotropic redistribution is dissipative.

###  II.4 Budget equation for GS Reynolds stress spectrum

For a further understanding of SGS modeling, an analysis of the energy budget
in scale space is useful. Domaradzki et al. [16] showed that in turbulent
channel flows, the local energy transfer in scale space through the nonlocal
wavenumber triad interaction is the majority in the interscale energy transfer
process. This result suggests that the interscale interaction across the
cutoff scale plays a significant role in energy transfer, which should be
implemented in SGS stress. As another approach to investigating interscale
interaction, Cimarelli and De Angelis [26] analyzed the Kolmogorov equation,
which is the budget equation for the second-order velocity structure function,
for both unfiltered and filtered velocity fields. They concluded that to
resolve the interscale interaction including the inverse cascade observed in
the spanwise scale, the appropriate filter length scale for eddy-viscosity
models yields $\overline{\Delta}_{x}^{+}<100$ and
$\overline{\Delta}_{z}^{+}<20$ where $\overline{\Delta}_{i}$ denotes the
filter length scale for the $i$-th direction, $x$ and $z$ are the streamwise
and spanwise directions, and the values with a superscript $+$ denote those
normalized by the wall shear stress and kinematic viscosity. In other words,
we have to employ additional stress to the eddy-viscosity term when using a
larger filter length in LES. This limitation of the filter length scale is
consistent with the typical grid resolution for conventional LES employing
eddy-viscosity models, $\overline{\Delta}_{x}^{+}<130$ and
$\overline{\Delta}_{z}^{+}<30$ (see Refs. [27, 28, 29]).

To investigate the physical role of anisotropic stress and the limitations of
eddy-viscosity models, we also analyze the budget equation for GS Reynolds
stress spectrum. We consider the case in which the $x$ and $z$ directions are
periodic and the $y$ direction is bounded by solid walls as turbulent channel
flows. We adopt the Fourier transformation in the $x$ and $z$ directions for
the scale decomposition of the flow fields; the discrete Fourier
transformation of a quantity $q$ and its inverse transformation read

$\displaystyle\tilde{q}(n_{x},y,n_{z})$
$\displaystyle=\frac{1}{N_{x}N_{z}}\sum_{(I,K)=(1,1)}^{(N_{x},N_{z})}q(x_{I},y,z_{K})\exp[-2\pi\mathrm{i}(n_{x}x_{I}/L_{x}+n_{z}z_{K}/L_{z})],$
(11a) $\displaystyle q(x_{I},y,z_{K})$
$\displaystyle=\sum_{(n_{x},n_{z})=(-N_{x}/2,-N_{z}/2)}^{(N_{x}/2-1,N_{z}/2-1)}\tilde{q}(n_{x},y,n_{z})\exp[2\pi\mathrm{i}(n_{x}x_{I}/L_{x}+n_{z}z_{K}/L_{z})],$
(11b)

where $\mathrm{i}=\sqrt{-1}$, $x_{I}=L_{x}I/N_{x}$, and $z_{K}=L_{z}K/N_{z}$.
$L_{i}$ and $N_{i}$ are the length of the flow domain and grid number in the
$i$-th direction, respectively. When the turbulence field is statistically
homogeneous in the $x$ and $z$ directions, the second-order correlation yields
$\langle f(x_{I},y,z_{K})g(x_{I},y,z_{K})\rangle=\langle
fg\rangle(y)=\sum_{n_{x},n_{z}}\Re\langle\tilde{f}(n_{x},y,n_{z})\tilde{g}^{*}(n_{x},y,n_{z})\rangle$
where the superscript $*$ represents the complex conjugate. Hence, the GS
Reynolds stress spectrum $E^{\mathrm{GS}}_{ij}(k_{x},y,k_{z})$ reads

$\displaystyle
E^{\mathrm{GS}}_{ij}(k_{x},y,k_{z})=\Re\left<\tilde{\overline{u}}_{i}^{\prime}\tilde{\overline{u}}_{i}^{\prime}{}^{*}\right>,$
(12)

which satisfies
$R^{\mathrm{GS}}_{ij}(y)=\sum_{n_{x},n_{z}}E^{\mathrm{GS}}_{ij}(k_{x},y,k_{z})$
where $k_{x}=2\pi n_{x}/L_{x}$ and $k_{z}=2\pi n_{z}/L_{z}$.

The extension of the spectral budget for Reynolds stress [30, 31] to the
filtered Navier-Stokes equations yields [8]

$\displaystyle\frac{\partial E^{\mathrm{GS}}_{ij}}{\partial
t}=\check{P}^{\mathrm{GS}}_{ij}-\check{\varepsilon}^{\mathrm{GS}}_{ij}+\check{\Phi}^{\mathrm{GS}}_{ij}+\check{D}^{\mathrm{p,GS}}_{ij}+\check{D}^{\mathrm{t,GS}}_{ij}+\check{D}^{\mathrm{v,GS}}_{ij}-\check{\varepsilon}^{\mathrm{SGS}}_{ij}+\check{D}^{\mathrm{SGS}}_{ij}+\check{T}^{\mathrm{GS}}_{ij},$
(13)

where

$\displaystyle\check{P}^{\mathrm{GS}}_{ij}$
$\displaystyle=-E^{\mathrm{GS}}_{iy}\frac{\partial U_{j}}{\partial
y}-E^{\mathrm{GS}}_{jy}\frac{\partial U_{i}}{\partial y},$ (14a)
$\displaystyle\check{\varepsilon}^{\mathrm{GS}}_{ij}$
$\displaystyle=2\nu\Re\left<\tilde{\overline{s}}_{i\ell}^{\prime}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{j}^{\prime})^{*}+\tilde{\overline{s}}_{j\ell}^{\prime}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{i}^{\prime})^{*}\right>,$
(14b) $\displaystyle\check{\Phi}^{\mathrm{GS}}_{ij}$
$\displaystyle=2\Re\left<\tilde{\overline{p}}^{\mathrm{total}}\tilde{\overline{s}}_{ij}^{\prime}{}^{*}\right>,$
(14c) $\displaystyle\check{D}^{\mathrm{p,GS}}_{ij}$
$\displaystyle=-\frac{\partial}{\partial
y}\Re\left<\tilde{\overline{p}}^{\mathrm{total}}\tilde{\overline{u}}_{j}^{\prime}{}^{*}\delta_{iy}+\tilde{\overline{p}}^{\mathrm{total}}\tilde{\overline{u}}_{j}^{\prime}{}^{*}\delta_{iy}\right>,$
(14d) $\displaystyle\check{D}^{\mathrm{t,GS}}_{ij}$
$\displaystyle=-\frac{1}{2}\frac{\partial}{\partial
y}\Re\left<\widetilde{\overline{u}_{y}^{\prime}\overline{u}_{i}^{\prime}}\tilde{\overline{u}}_{j}^{\prime}{}^{*}+\widetilde{\overline{u}_{y}^{\prime}\overline{u}_{j}^{\prime}}\tilde{\overline{u}}_{i}^{\prime}{}^{*}\right>,$
(14e) $\displaystyle\check{D}^{\mathrm{v,GS}}_{ij}$
$\displaystyle=2\nu\frac{\partial}{\partial
x_{y}}\Re\left<\tilde{\overline{s}}_{iy}^{\prime}\tilde{\overline{u}}_{j}^{\prime}{}^{*}+\tilde{\overline{s}}_{jy}^{\prime}\tilde{\overline{u}}_{j}^{\prime}{}^{*}\right>,$
(14f) $\displaystyle\check{\varepsilon}^{\mathrm{SGS}}_{ij}$
$\displaystyle=-\Re\left<\tilde{\tau}^{\mathrm{sgs}}_{i\ell}|_{\mathrm{tl}}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{j}^{\prime})^{*}+\tilde{\tau}^{\mathrm{sgs}}_{j\ell}|_{\mathrm{tl}}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{i}^{\prime})^{*}\right>,$
(14g) $\displaystyle\check{D}^{\mathrm{SGS}}_{ij}$
$\displaystyle=-\frac{\partial}{\partial
y}\Re\left<\tilde{\tau}^{\mathrm{sgs}}_{iy}|_{\mathrm{tl}}\tilde{\overline{u}}_{j}^{\prime}{}^{*}+\tilde{\tau}^{\mathrm{sgs}}_{j\ell}|_{\mathrm{tl}}\tilde{\overline{u}}_{i}^{\prime}{}^{*}\right>,$
(14h) $\displaystyle\check{T}^{\mathrm{GS}}_{ij}$
$\displaystyle=\Re\left<\tilde{N}_{i}\tilde{\overline{u}}_{j}^{\prime}{}^{*}+\tilde{N}_{j}\tilde{\overline{u}}_{i}^{\prime}{}^{*}\right>-\check{P}^{\mathrm{GS}}_{ij}-\check{D}^{\mathrm{t,GS}}_{ij},$
(14i)

$(\tilde{\partial}_{x},\tilde{\partial}_{y},\tilde{\partial}_{z})=(\mathrm{i}k_{x},\partial/\partial
y,\mathrm{i}k_{z})$, and
$N_{i}=-\partial\overline{u}_{i}\overline{u}_{\ell}/\partial x_{\ell}$. Here,
we assumed homogeneity of the turbulence field in the $x$ and $z$ directions.
All terms on the right-hand side of Eq (13) except for
$\check{T}^{\mathrm{GS}}_{ij}$ lead to the right-hand side of Eq. (6) when
summed over the wavenumber; that is,

$\displaystyle
A_{ij}(y)=\sum_{(n_{x},n_{z})=(-N_{x}/2,-N_{z}/2)}^{(N_{x}/2-1,N_{z}/2-1)}\check{A}_{ij}(k_{x},y,k_{z}),$
(15)

for
$A_{ij}=(P^{\mathrm{GS}}_{ij},\varepsilon^{\mathrm{GS}}_{ij},\Phi^{\mathrm{GS}}_{ij},D^{\mathrm{p,GS}}_{ij},D^{\mathrm{t,GS}}_{ij},D^{\mathrm{v,GS}}_{ij},\varepsilon^{\mathrm{SGS}}_{ij},D^{\mathrm{SGS}}_{ij})$.
We refer to these terms with the same names as those given in Eqs. (7a)–(7h).
However, the sum of $\check{T}^{\mathrm{GS}}_{ij}$ over the wavenumber yields
zero; that is,

$\displaystyle\sum_{(n_{x},n_{z})=(-N_{x}/2,-N_{z}/2)}^{(N_{x}/2-1,N_{z}/2-1)}\check{T}_{ij}(k_{x},y,k_{z})=0.$
(16)

Therefore, $\check{T}^{\mathrm{GS}}_{ij}$ represents the transfer of
$E^{\mathrm{GS}}_{ij}$ among the wavenumber modes. We refer to
$\check{T}^{\mathrm{GS}}_{ij}$ as the interscale transfer term.

For the spectral expressions of SGS dissipation, we also employ the
decomposition provided in Sec. II.3. Namely,

$\displaystyle\check{\varepsilon}^{\mathrm{SGS}}_{ij}$
$\displaystyle=\check{\varepsilon}^{\mathrm{EV}}_{ij}-\check{\xi}^{\mathrm{AR}}_{ij},$
(17)

where

$\displaystyle\check{\varepsilon}^{\mathrm{EV}}_{ij}$
$\displaystyle=2\left<\widetilde{\nu^{\mathrm{sgs}}\overline{s}_{i\ell}}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{j}^{\prime})^{*}+\widetilde{\nu^{\mathrm{sgs}}\overline{s}_{j\ell}}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{i}^{\prime})^{*}\right>,$
(18a) $\displaystyle\check{\xi}^{\mathrm{AR}}_{ij}$
$\displaystyle=\left<\tilde{\tau}^{\mathrm{ani}}_{i\ell}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{j}^{\prime})^{*}+\tilde{\tau}^{\mathrm{ani}}_{j\ell}(\tilde{\partial}_{\ell}\tilde{\overline{u}}_{i}^{\prime})^{*}\right>.$
(18b)

Both terms on the right-hand side of Eqs. (17) obey Eq. (15) and are referred
to as eddy-viscosity dissipation and anisotropic redistribution, respectively.

##  III Results in turbulent channel flows

###  III.1 Numerical setup

We investigate the budget equation for GS Reynolds stress provided in Sec.
II.2 using the DNS data of the turbulent channel flows. The $x$, $y$, and $z$
coordinates represent the streamwise, wall-normal, and spanwise directions,
respectively. We employed a staggered grid system and adopted a fourth-order
conservative central finite difference scheme for the $x$ and $z$ directions
and a second-order conservative finite difference scheme on the non-uniform
grid for the $y$ direction [32]. Periodic boundary conditions were employed in
the $x$ and $z$ directions and no-slip condition was employed in the $y$
direction. The second-order Adams-Bashforth scheme was employed, for time
marching. The Poisson equation for pressure was solved using a fast Fourier
transformation. The Reynolds number is set to be $\mathrm{Re}_{\tau}=400$
where $\mathrm{Re}_{\tau}=u_{\tau}h/\nu$, $u_{\tau}(=\sqrt{\nu|\partial
U_{x}/\partial y|_{\mathrm{wall}}|})$ is the friction velocity, and $h$ is the
channel half width. The computational domain size is $L_{x}\times L_{y}\times
L_{z}=2\pi h\times 2h\times\pi h$ and the number of grid is $N_{x}\times
N_{y}\times N_{z}=256\times 192\times 256$. The numerical resolutions in each
direction are $\Delta x^{+}=9.8$, $\Delta y^{+}=0.34$-$10$, and $\Delta
z^{+}=4.9$, respectively. Here and hereafter, the values with a superscript
$+$ denote those normalized by $u_{\tau}$ and $\nu$. The statistical average
is obtained over the $x$-$z$ plane and time. The validation of our simulation
is discussed in Appendix B.

To calculate the filtered quantities, we adopted a Fourier sharp-cut filter,
which is commutative with a differential operation. Note that in the calculus
of GS and SGS dissipation spectra, Eqs. (14b) and (14g), we have to use the
modified wavenumber for $\tilde{\partial}_{i}$ owing to employing a finite
difference scheme [32]; namely, for the fourth-order central finite
difference, $k_{x}^{\mathrm{mod}}=[27\sin(\pi n_{x}/N_{x})-\sin(3\pi
n_{x}/N_{x})]/(12\Delta x)$. For simplicity, the filter operation is applied
only in the $x$ and $z$ directions. In the actual LES or a posteriori tests,
the filter in the wall-normal direction is often applied because the grid size
along the wall-normal direction is coarser than that of DNS. Furthermore, the
scale of eddies in the wall-normal direction is also important to discuss the
dynamics of wall-bounded turbulent flows [33]. Thus, the filter in the wall-
normal direction can provide physical insight into wall-bounded turbulent
flows. However, if we realize the filter that coincides with the actual LES,
we have to apply the inhomogeneous filter where the filter length changes
against the distance from the wall. This inhomogeneous filter is not
commutative with a differential operation and induces several additional terms
which are different from the stress term in the filtered continuity and
Navier-Stokes equations [14]. In the conventional LES, we often ignore such
terms arising from the commutation error and model only the SGS stress or
related numerical viscosity. In this study, we focus on the role of SGS stress
in the budget for GS Reynolds stress and its spectrum, which should be
reproduced in the actual LES. For this reason, we apply the filter only in the
$x$ and $z$ directions. When the filter is applied only in homogeneous
directions, the statistical average of the filtered variable yields the same
as that of the unfiltered variable, namely,
$\langle\overline{q}\rangle=\langle q\rangle$. Owing to this property, the
second moment can be straightforwardly decomposed into three of mean, GS
turbulence, and SGS parts. For example, the statistical average of kinetic
energy $\langle u_{i}u_{i}\rangle/2$ is decomposed into mean $U_{i}U_{i}/2$,
GS turbulence
$K^{\mathrm{GS}}(=\langle\overline{u}_{i}^{\prime}\overline{u}_{i}^{\prime}\rangle/2)$,
and SGS
$\langle\overline{u_{i}u_{i}}-\overline{u}_{i}\overline{u}_{i}\rangle/2$. If
we also apply the filtering in the wall-normal direction, we have
$\langle\overline{q}\rangle\neq\langle q\rangle$, and thus the mean of
filtered velocity $\langle\overline{u}_{x}\rangle$ is somewhat smoothed
compared with the mean velocity of DNS $\langle u_{x}\rangle$. However, in the
near-wall to buffer region, LES often predicts the mean velocity profile well.
Therefore, we infer that the interaction between the mean flow and GS
turbulent field through the mean velocity gradient can be reasonably discussed
to some extent even if we apply the filter only in the $x$ and $z$ directions.
To observe the filter-size dependence, we employed two filter sets, namely,
medium-size filter where the set of cutoff wavelength scales is
$(\lambda^{\mathrm{c}+}_{x},\lambda^{\mathrm{c}+}_{z})=(105,52.4)$ and coarse-
or large-size filter where
$(\lambda^{\mathrm{c}+}_{x},\lambda^{\mathrm{c}+}_{z})=(209,105)$. Here
$\lambda_{\alpha}=2\pi/k_{\alpha}$ for $\alpha=x,z$ that correspond to LES
resolutions of $(\Delta x^{+},\Delta z^{+})=(52.4,26.2)$ and $(105,52.4)$,
respectively. Strictly speaking, an exact cut-off wavelength is often slightly
shorter than the length of grid size because of dealiasing in a spectral
scheme or the accuracy of the discretization scheme. For example for a fourth-
order central finite difference scheme, spatial differentiation yields the
modified wavenumber mentioned above whose modulus is smaller than the exact
wavenumber. In this study, we do not consider such an effective maximum
wavenumber to simplify the interpretation of the analysis. According to the
typical resolutions referred to in Sec. II.4, the eddy-viscosity models can
predict the basic statistics for the medium filter case, whereas the large
filter case is outside their range (see also Appendix C). Hence, by comparing
the results of these two filter sizes, we can determine the physical role of
the anisotropic SGS stress.

###  III.2 Filter wavelength and GS Reynolds stresses

Figure 1: Profiles of GS Reynolds stress for (a) streamwise, (b) wall-normal,
(c) spanwise, and (d) shear components. The black line with crosses depicts
the result of the DNS without filter operation.

Figure 1 shows the profiles of each non-zero component of GS Reynolds stress.
The profiles for DNS without filter operation are also plotted. Most of the
turbulent velocity fluctuations are resolved in the GS for the medium filter
case. The ratio of
$K^{\mathrm{GS}}(=\langle\overline{u}_{i}^{\prime}\overline{u}_{i}^{\prime}\rangle/2=R^{\mathrm{GS}}_{ii}/2)$
to the total turbulent kinetic energy $K(=\langle
u_{i}^{\prime}u_{i}^{\prime}\rangle/2=K^{\mathrm{GS}}+\langle\tau^{\mathrm{sgs}}_{ii}\rangle/2))$
yields approximately greater than 90% over the entire region
$K^{\mathrm{GS}}/K\gtrsim 0.9$, which is within the tolerance of the
conventional LES [34]. However, for the coarse filter case, the ratio
$K^{\mathrm{GS}}/K$ is less than 80% in $y^{+}<100$. Hence, the conventional
eddy-viscosity-based LES with a coarse grid fails to predict the statistics of
channel flows.

Figure 2: Pre-multiplied Reynolds stress spectrum
$k_{x}k_{z}E_{ij}(\lambda_{x},\lambda_{z})$ where
$E_{ij}=\Re\langle\tilde{u}_{i}^{\prime}\tilde{u}_{j}^{\prime}{}^{*}\rangle$,
which is defined by the unfiltered velocity field. (a,b) Streamwise, (c,d)
wall-normal, (e,f) spanwise, and (g,h) shear components. Wall-normal heights
for (a,c,e,g) and (b,d,f,h) are $y^{+}\approx 15$ and $y^{+}\approx 100$,
respectively. The solid and dashed lines depict the medium and coarse filter
length scales, respectively.

To understand the physical meaning of the filter length, two-dimensional pre-
multiplied spectra of the Reynolds stress are shown in Fig. 2. Here and
hereafter, $k_{x}$ and $k_{z}$ denote their moduli when considering the
statistical values. We chose two heights: the near-wall region $y^{+}\approx
15$ and slightly away from the wall $y^{+}\approx 100(y/h\approx 0.25)$. For
the medium filter, almost all the components are well-resolved. The wall-
normal component is partially filtered out in the near-wall region even for
the medium filter. The ratio of wall-normal GS velocity fluctuation to the
unfiltered one is less than 80% in $y^{+}<50$ for the medium filter case. This
may be one of the reasons why the wall-normal velocity fluctuation is often
underestimated in LESs (see Appendix C). To predict the anisotropy in the
near-wall region of shear flows more accurately, a finer grid may be required
in the spanwise direction, for example, $\Delta z^{+}<20$. The length scale
$\Delta z^{+}=20$ is comparable to that specified by the inverse energy
transfer in the spanwise scale proposed by Cimarelli and De Angelis [26].

For the coarse filter, half of the peak of the streamwise spectrum is filtered
out in the near-wall region $y^{+}\approx 15$. This indicates that the filter
length scale lies within the energy-containing scale. In addition, most of the
wall-normal and shear stress spectra are contained in the SGS. Therefore,
conventional isotropic eddy-viscosity models are not valid for coarse filters.
Furthermore, in the region slightly away from the wall $y^{+}\approx 100$, the
wall-normal and spanwise spectra are partially filtered out, whereas the
streamwise and shear stress spectra are well-resolved. Hence, the SGS
anisotropy is significant even in the region slightly away from the wall in
the coarse filter case.

###  III.3 Mean SGS shear stress

Figure 3: Profiles of SGS shear stresses. The solid and dashed lines represent
the medium and coarse filter cases, respectively. For both filter sizes, the
black, red, and blue lines depict the total SGS stress
$-\langle\tau^{\mathrm{sgs}}_{xy}\rangle$, eddy-viscosity term
$2\langle\nu^{\mathrm{sgs}}\overline{s}_{xy}\rangle$, and anisotropic term
$-\langle\tau^{\mathrm{ani}}_{xy}\rangle$ given by Eq. (5), respectively. Here
and hereafter, the light-gray dotted line is the zero line.

It is important to verify the mean SGS shear stress profile. In general, it is
difficult to reproduce both the energy transfer rate and mean SGS shear stress
using only the eddy-viscosity term [1, 2, 3]. Because the eddy viscosity in
this study is determined to reproduce the energy transfer rate, it does not
necessarily predict the accurate mean SGS shear stress. Figure 3 shows the
profiles of mean SGS shear stress in terms of the SGS stress decomposition.
For the medium filter, the contribution of anisotropic stress is negligible.
Therefore, the eddy-viscosity models can predict both the energy transfer rate
and mean SGS shear stress for the medium filter case. This result provides an
interpretation of why LES with an eddy-viscosity model works well for the
medium grid resolution. In contrast, for the coarse filter, anisotropic stress
is comparable to the eddy-viscosity term. Hence, the anisotropic stress is
necessary for the coarse filter case to predict both the energy transfer rate
and mean SGS shear stress.

###  III.4 Budget for GS turbulent kinetic energy

Figure 4: Budget for the GS turbulent kinetic energy $K^{\mathrm{GS}}$ for (a)
medium and (b) coarse filter cases. Here and hereafter, the black dashed line
depicts the residual or sum of all terms for the budgets.

Figure 4 shows the budget for GS turbulent kinetic energy $K^{\mathrm{GS}}$.
Each term corresponds to half of the trace of Eqs. (7a)–(7f), (9a), (9b), and
(7h). Here and hereafter, we only show $y^{+}\leq 200$ for the budgets to
highlight the near-wall features. The trend of each term of the budget is
almost monotonic in $y^{+}>200$. The pressure redistribution (7c) is not
plotted because it is traceless. For the conventional unfiltered turbulent
kinetic energy budget, the production almost balances the dissipation in the
region away from the wall $y^{+}>30$ (see e.g. Ref. [35]). The eddy-viscosity
dissipation term also contributes to the budget as an energy sink for the GS
turbulent kinetic energy budget. As expected, the contribution of eddy-
viscosity dissipation becomes dominant for the coarse filter. The anisotropic
redistribution term is negligible for the medium filter. In contrast, this
term has a positive value in the near-wall region for the coarse filter. These
results are consistent with the profiles of
$\langle\tau^{\mathrm{ani}}_{xy}\rangle$ shown in Fig. 3. Because the trace of
the anisotropic redistribution is given by Eq. (10), it becomes prominent when
the mean anisotropic stress $\langle\tau^{\mathrm{ani}}_{xy}\rangle$
increases. Note that the SGS dissipation, which is the sum of eddy-viscosity
dissipation and anisotropic redistribution, is positive at $y^{+}\approx 12$
for the coarse filter. Furthermore, the eddy-viscosity dissipation is also
positive at $y^{+}\approx 10$, although its value is much smaller than that of
SGS dissipation. The productive contribution or averaged backward energy
transfer of SGS dissipation in the near-wall region has already been discussed
[25, 16, 17, 36, 26]. Several studies suggested a relationship between
backward energy transfer and coherent structures observed in the near-wall
region of turbulent shear flows [25, 37, 26, 38]. We discuss this in Sec.
IV.1.

Figure 5: Profiles of the normal components of anisotropic redistribution
$\xi^{\mathrm{GS}}_{\alpha\alpha}$ where summation is not taken for
$\alpha(=x,y,z)$ for (a) medium and (b) coarse filter cases. The black line
depicts the trace part $\xi^{\mathrm{AR}}_{ii}$.

The anisotropic stress is not necessarily negligible even when the trace of
the anisotropic redistribution term is small when compared with other terms.
Figure 5 shows the profiles of the normal components of anisotropic
redistribution. For the medium filter, the wall-normal and spanwise components
are almost canceled out. For the coarse filter, the positive contributions of
the streamwise and spanwise components increase at $y^{+}\approx 20$. These
two components contribute positively to the GS turbulent kinetic energy budget
in the near-wall region as shown in Fig. 4(b). In addition, the spanwise
component is dominant in $50<y^{+}<100$ for the coarse filter. Notably, the
spanwise component of the anisotropic redistribution is always positive for
both filter sizes. In turbulent channel flows, the production term is zero in
the budget for the spanwise component of GS Reynolds stress
$R^{\mathrm{GS}}_{zz}$; that is, $P^{\mathrm{GS}}_{zz}=0$. The productive
term, which is the pressure redistribution in the conventional unfiltered
budget, plays a significant role in the budget for spanwise GS Reynolds
stress. In the next subsection, we examine the budget for the normal
components of GS Reynolds stress.

###  III.5 Budget for GS Reynolds stress

In this study, we focus on the budget only for the normal components of GS
Reynolds stress. A detailed analysis of the contributions of anisotropic SGS
stress to the shear stress budget was already demonstrated by Abe [13], which
concluded that anisotropic SGS stress is essential to reproduce the productive
contribution to the shear stress budget.

####  III.5.1 Streamwise component

Figure 6: Budget for the streamwise component of GS Reynolds stress
$R^{\mathrm{GS}}_{xx}$ for (a) medium and (b) coarse filter cases.

Figure 6 shows the budget for the streamwise component of GS Reynolds stress.
The basic profiles of each term are almost the same as those of the GS
turbulent kinetic energy budget shown in Fig. 4. The pressure redistribution
term, which transfers the energy from the streamwise component to the other
two components, is an exception. The intensity of the pressure redistribution
is small when compared to the eddy-viscosity dissipation for the coarse
filter. However, this does not imply that the redistribution among the normal
components is negligible because pressure redistribution is an essential
source term in the wall-normal and spanwise components.

####  III.5.2 Wall-normal component

Figure 7: Budget for the wall-normal component of GS Reynolds stress
$R^{\mathrm{GS}}_{yy}$ for (a) medium and (b) coarse filter cases.

Figure 7 shows the budget for the wall-normal component of GS Reynolds stress.
The pressure redistribution has a large productive contribution except for the
close vicinity of the wall $y^{+}<10$. The viscous and eddy-viscosity
dissipations are the leading terms of the negative contribution for both
filter sizes. For the coarse filter, the eddy-viscosity dissipation is
dominant, as seen in the budget for the streamwise component. In the near-wall
to buffer region $y^{+}<100$, the SGS diffusion term also contributes
significantly to the budget and plays a key role in counterbalancing the sum
of the pressure and turbulent diffusions. The details of SGS diffusion are
provided in Appendix A. The anisotropic redistribution has a large negative
contribution compared to the viscous and eddy-viscosity dissipations in the
near-wall region $y^{+}<50$. Conventional eddy-viscosity models may account
for this negative contribution in the a posteriori tests, although the
intensity could be small. Thus, the leading productive and dissipative
contributions to the wall-normal GS Reynolds stress can be reproduced by the
pressure redistribution, viscous dissipation, and eddy-viscosity dissipation
terms.

####  III.5.3 Spanwise component

Figure 8: Budget for the spanwise component of GS Reynolds stress
$R^{\mathrm{GS}}_{zz}$ for (a) medium and (b) coarse filter cases. We also
plot the sum of eddy-viscosity dissipation and anisotropic redistribution,
which is depicted as SGS dissipation in the purple dashed line with inverted
triangles.

Figure 8 shows the budget for the spanwise component of GS Reynolds stress.
The pressure redistribution has a leading productive contribution similar to
the wall-normal component shown in Fig. 7. In addition, the viscous and eddy-
viscosity dissipations are also the leading terms of the negative
contribution. An important finding is that the anisotropic redistribution is
always positive in the spanwise GS Reynolds stress budget as shown in Fig. 5.
Furthermore, the sum of eddy-viscosity dissipation and anisotropic
redistribution is positive near the wall $y^{+}\approx 20$ for both filter
sizes. The intensity of the anisotropic redistribution is relatively small
compared to that of the pressure redistribution for the medium filter. In
contrast, the anisotropic redistribution is comparable to the pressure
redistribution in the near-wall to buffer region $y^{+}<100$ for the coarse
filter. Therefore, this positive contribution of anisotropic redistribution is
indispensable to the generation mechanism of the spanwise velocity fluctuation
in the GS or the resolved scale for the coarse filter case. Even for the
medium filter, we infer that the lack of anisotropic redistribution will lead
to an underestimation of the GS spanwise velocity fluctuation (see also
Appendix C). The underestimation of GS spanwise velocity fluctuation may alter
the structure of the wall-bounded turbulent shear flows, for example, coherent
structures in the near-wall region. Hamba [38] demonstrated that the
conditional averaged velocity field regarding the inverse cascade of the
spanwise velocity fluctuation represents the streamwise elongated vorticity
structure, which represents the coherent structure in wall-bounded turbulent
shear flows. A relationship between backward scatter in terms of kinetic
energy and coherent structures has also been suggested [25, 37, 26]. The
present analysis suggests that the anisotropic SGS stress reproducing the
productive contribution to the spanwise GS Reynolds stress budget is key to
improving SGS models.

##  IV Discussion

In Sec. III.5, we demonstrated that the anisotropic redistribution term
contributes positively to the streamwise and spanwise components of GS
Reynolds stress. Conventional eddy-viscosity models cannot represent the
productive contribution. In this section, first, we discuss the physical
interpretation of the productive contribution in terms of the budget equation
for GS Reynolds stress spectrum. Second, we perform an a priori test of the
anisotropic redistribution term based on several existing model expressions.

###  IV.1 Budget for GS Reynolds stress spectrum

Several studies have discussed the relationship between inverse cascade and
coherent structures in wall-bounded turbulent shear flows [25, 37, 26, 38].
Spectral analysis is a fundamental tool used to study the effect of each term
in the budget on structures represented by specific scales [30, 31]. A
representative of the near-wall structure is streaky structures whose spanwise
spacing or wavelength is $\lambda_{z}^{+}\sim 100$ [39, 40]. According to the
self-sustaining process of wall-bounded turbulent shear flows, streaky
structures are generated by streamwise vortices, and the breakdown of the
streaks generates the source modes of the nonlinear interaction that generates
the streamwise vortices [41, 42]. In particular, the generation of streamwise
vortices does not agree with the dissipative property of eddy viscosity [38].
To determine the relationship between these processes and the productive
contribution of anisotropic redistribution in the streamwise and spanwise
components of GS Reynolds stress, we examine the budget for
$E^{\mathrm{GS}}_{xx}(k_{x})$ and $E^{\mathrm{GS}}_{zz}(k_{z})$. Here and
hereafter, we simply denote
$E^{\mathrm{GS}}_{ij}(k_{x})=\sum_{n_{z}}E^{\mathrm{GS}}_{ij}(k_{x},k_{z})$ or
$E^{\mathrm{GS}}_{ij}(k_{z})=\sum_{n_{x}}E^{\mathrm{GS}}_{ij}(k_{x},k_{z})$.
Although $E^{\mathrm{GS}}_{zz}(k_{z})$ itself does not directly represent the
vorticity, the spanwise velocity fluctuation accompanied by the nonzero
spanwise wavenumber is related to the streamwise vortices as
$\tilde{\omega}_{x}(k_{z})=\partial\tilde{u}_{z}(k_{z})/\partial
y-\mathrm{i}k_{z}\tilde{u}_{y}(k_{z})$.

####  IV.1.1 Streamwise component

Figure 9: Budget for the streamwise component of GS Reynolds stress spectrum
in the streamwise wavenumber space $E^{\mathrm{GS}}_{xx}(k_{x})$ for (a)
medium and (b) coarse filter cases at $y^{+}\approx 15$. The insets show the
small vertical axis range to focus on the small-scale or high-wavenumber
region.

Figure 9 shows the budget for the streamwise component of GS Reynolds stress
spectrum in the streamwise wavenumber space $E^{\mathrm{GS}}_{xx}(k_{x})$ at
$y^{+}\approx 15$. We plot the budget with a linear scale in $k_{x}$ instead
of a log scale or wavelength scale to depict the $k_{x}=0$ mode. For both
filter sizes, the production term contributes significantly in the low-
wavenumber region. For the medium filter, the gain by the interscale
interaction is balanced with the loss by viscous dissipation and pressure
redistribution in the high-wavenumber region. In contrast, for the coarse
filter, the anisotropic redistribution has a positive contribution in the
high-wavenumber region $k_{x}^{+}>0.01$ ($\lambda_{x}^{+}\lesssim 600$). The
contribution of anisotropic redistribution to the high-wavenumber mode was
already highlighted by Inagaki and Kobayashi [8] in the low-Reynolds number
case. They suggested that in coarse LES using only the eddy-viscosity term,
the absence of enhancement of small scales causes a longer streamwise velocity
correlation in the $x$ direction than that of filtered DNS. In other words,
the anisotropic term other than eddy viscosity is needed in coarse LES to
predict the streamwise velocity correlation in the $x$ direction comparable to
that of filtered DNS.

Figure 10: Two-dimensional spectra of (a) the interscale transfer
$\check{T}^{\mathrm{GS}}_{xx}(k_{x},k_{z})$ and (b) anisotropic redistribution
$\check{\xi}^{\mathrm{AR}}_{xx}(k_{x},k_{z})$ for the coarse filter case at
$y^{+}\approx 15$.

To see the details of the gain in the small scales, we examine the two-
dimensional spectra of the interscale transfer and anisotropic redistribution
in the $k_{x}$-$k_{z}$ plane for the coarse filter case. Figure 10 shows the
contributions of interscale transfer and anisotropic redistribution at
$y^{+}\approx 15$. The interscale transfer term transports
$E^{\mathrm{GS}}_{xx}(k_{x},k_{z})$ from the region $k_{x}^{+}<0.01$ and
$0.02<k_{z}^{+}<0.06$ ($\lambda_{x}^{+}\gtrsim 600$ and
$100\lesssim\lambda_{z}^{+}\lesssim 300$) to the modes with a relatively large
spanwise length scale where $k_{z}^{+}<0.03$ ($\lambda_{z}^{+}\gtrsim 200$).
In contrast, the anisotropic redistribution term contributes to the high-
wavenumber region $k_{x}^{+}>0.015$ ($\lambda_{x}^{+}\lesssim 400$).
Therefore, we can interpret that the anisotropic redistribution plays a key
role in the amplification of small-scale mode in the streamwise scale. The
typical spanwise length scale of streaky structures lies on a scale close to
the cutoff $\lambda_{z}^{+}\sim 100\sim\lambda_{z}^{\mathrm{c}+}$ for the
coarse filter. Focusing on $k_{x}^{+}>0.015$ and $k_{z}^{+}>0.04$
($\lambda_{x}^{+}\lesssim 400$ and $\lambda_{z}^{+}\lesssim 160$), the
anisotropic redistribution has a slightly larger contribution to the budget
than interscale transfer. This generation of a small streamwise length scale
mode with $\lambda_{z}^{+}\sim 100$ is consistent with the streak breakdown.
We infer that this positive contribution of anisotropic redistribution can
improve the performance of LES in the way that it reproduces the streak
breakdown-like contribution.

####  IV.1.2 Spanwise component

Figure 11: Budget for the spanwise component of GS Reynolds stress spectrum in
the spanwise wavenumber space $E^{\mathrm{GS}}_{zz}(k_{z})$ for (a) medium and
(b) coarse filter cases at $y^{+}\approx 15$.

Figure 11 shows the budget for the spanwise component of GS Reynolds stress
spectrum in the spanwise wavenumber space $E^{\mathrm{GS}}_{zz}(k_{z})$ at
$y^{+}\approx 15$. The interscale transfer is negative at $k_{z}^{+}=0.04$ and
positive in $k_{z}^{+}<0.02$ for both filter sizes, which represents the
inverse transfer of $E^{\mathrm{GS}}_{zz}(k_{z})$ in the spanwise scale. The
inverse transfer of the spanwise velocity fluctuation has already been
identified by Hamba [38], although it has been demonstrated in the streamwise
scale space. In addition, the inverse transfer of kinetic energy in the
spanwise scale has already been discussed [26, 43]. For the medium filter, the
interscale transfer changes the sign at $k_{z}^{+}\approx 0.08$
($\lambda_{z}^{+}\approx 80$) in the high-wavenumber region. This critical
length scale is larger than that observed in the analysis based on the
Kolmogorov equation [26] or the scale energy density in terms of a filter
function [43]. They suggested that the critical length is $r_{z}^{+}\approx
20$ where $r_{z}$ denotes the distance between the two velocity fields
composing the scale. Note that the wavelength should be twice the distance
between the two velocity fields; that is $\lambda_{z}=2r_{z}$. The shift in
the critical length of interscale transfer is caused by the absence of small
scales, owing to the filtering operation. Nevertheless, we can infer that a
large amount of the interscale interaction in the budget of
$E^{\mathrm{GS}}_{zz}(k_{z})$ including inverse transfer is resolved in the
medium filter case. In contrast, for the coarse filter, the forward cascade of
$E^{\mathrm{GS}}_{zz}(k_{z})$ is completely unresolved. The unresolved
interscale interaction should be converted to the
$\tau^{\mathrm{sgs}}_{\ell\ell}$-related part of the pressure redistribution,
eddy-viscosity dissipation, and anisotropic redistribution. Part of the
forward cascade may convert to eddy-viscosity dissipation. However, for both
filter sizes, the positive contribution of anisotropic redistribution is
larger than the loss via eddy-viscosity dissipation in the entire wavenumber.
In addition, the anisotropic redistribution is comparable to the pressure
redistribution for the coarse filter.

Figure 12: Two-dimensional spectra of (a,b) the pressure redistribution
$\check{\Phi}^{\mathrm{GS}}_{zz}(k_{x},k_{z})$ and (c,d) anisotropic
redistribution $\check{\xi}^{\mathrm{AR}}_{zz}(k_{x},k_{z})$ for (a,c) medium
and (b,d) coarse filter cases at $y^{+}\approx 15$.

To observe the relationship between the coherent structures and positive
contributions to the budget, we examine the two-dimensional spectra of the
pressure and anisotropic redistributions in the $k_{x}$-$k_{z}$ plane. Figure
12 shows the two-dimensional spectra of the pressure and anisotropic
redistributions at $y^{+}\approx 15$. The pressure redistribution is negative
in the region $k_{x}^{+}<0.005$ and $k_{z}^{+}>0.04$ ($\lambda_{x}^{+}\gtrsim
1200$ and $\lambda_{z}^{+}\lesssim 160$) for both filter sizes. This region is
consistent with the scale of streamwise vortices in the self-sustaining
process [41, 42]. Because $E^{\mathrm{GS}}_{zz}(k_{z})$ can be related to the
streamwise vortices, we infer that the pressure redistribution attenuates the
streamwise vortices. The anisotropic redistribution is positive in this region
and peaks at $k_{z}^{+}\approx 0.05$ ($\lambda_{z}^{+}\approx 130$), which is
consistent with the typical spanwise spacing of streaks $\lambda_{z}^{+}\sim
100$. Therefore, we infer that the positive contribution of anisotropic
redistribution to the spanwise GS Reynolds stress spectrum is related to the
generation of streamwise vortices in the self-sustaining process [41, 42]. In
other words, the generation mechanism of streamwise vortices through the SGS
stress vanishes if we employ only the eddy-viscosity model in the LES. As we
can see from Fig. 12(c), the anisotropic redistribution has a large value at a
low-$k_{x}$ region, whereas it is large even at a relatively high-$k_{z}$
region. Therefore, we can infer that the filter in the $z$ direction is more
critical than that in the $x$ direction for the anisotropic redistribution.
The anisotropic redistribution is a significant source term at this scale even
in the medium filter case. Hence, by employing a proper anisotropic SGS stress
reproducing the productive contribution to the spanwise GS Reynolds stress,
the prediction of the statistics in LES for the coarse-to-medium filter cases
can be improved.

###  IV.2 A priori test of anisotropic redistribution term

A classical idea for implementing backward scatter in LES is stochastic
modeling [44]. Langford and Moser [45] demonstrated that the force from SGS
stress is mostly stochastic in their analysis of homogeneous isotropic
turbulence. They suggested that the SGS model may only be able to estimate the
average energy transfer rate from GS to SGS because of its stochastic nature.
However, the stochastic approach cannot predict the productive contribution to
the GS Reynolds stress budget; that is, even if we add the stochastic forcing
term $a_{i}$ to the filtered Navier-Stokes equations (2), its contribution to
the budget is always zero $\langle\overline{u}_{i}^{\prime}a_{j}\rangle=0$
because of its stochastic nature. Therefore, a deterministic model must be
employed to reproduce the positive contribution to the GS Reynolds stress
budget.

We examine the performance of several model expressions for the anisotropic
SGS stress in terms of an a priori test of the anisotropic redistribution. In
this study, the anisotropic stress is assumed to have no contribution to the
energy transfer between GS and SGS, according to Abe [6, 13]. Similarly, we
examine fundamental models based on the strategy proposed by Abe [6]. Namely,
we adopt the following two models for anisotropic stress:

$\displaystyle\tau^{\mathrm{ani}}_{ij}$
$\displaystyle=\tau^{\mathrm{sgs}}_{\ell\ell}\frac{\tau^{\mathrm{a}}_{ij}|_{\mathrm{tl}}+2\nu^{\mathrm{a}}\overline{s}_{ij}}{\tau^{\mathrm{a}}_{mm}},\
\
\tau^{\mathrm{a}}_{ij}=(\overline{u}_{i}-\widehat{\overline{u}}_{i})(\overline{u}_{j}-\widehat{\overline{u}}_{j}),$
(19) $\displaystyle\tau^{\mathrm{ani}}_{ij}$
$\displaystyle=\tau^{\mathrm{sgs}}_{\ell\ell}\frac{\tau^{\mathrm{a}}_{ij}|_{\mathrm{tl}}+2\nu^{\mathrm{a}}\overline{s}_{ij}}{\tau^{\mathrm{a}}_{mm}},\
\
\tau^{\mathrm{a}}_{ij}=\sum_{\alpha=1,2,3}\overline{\Delta}_{\alpha}^{2}\frac{\partial\overline{u}_{i}}{\partial
x_{\alpha}}\frac{\partial\overline{u}_{j}}{\partial x_{\alpha}},$ (20)

where
$\nu^{\mathrm{a}}=-\tau^{\mathrm{a}}_{ij}\overline{s}_{ij}/(2\overline{s}^{2})$
for both models, which is introduced to remove energy transfer through
$\tau^{\mathrm{a}}_{ij}$. The first model (19) is the scale-similarity model
for the SGS Reynolds term
$\overline{(u_{i}-\overline{u}_{i})(u_{j}-\overline{u}_{j})}\simeq(\overline{u}_{i}-\overline{\overline{u}}_{i})(\overline{u}_{j}-\overline{\overline{u}}_{j})$
[23], although the repeated filter is replaced with the test filter denoted by
$\widehat{\cdot}$. The filter length for the test filter is set to twice that
of the filter $\overline{\cdot}$; namely,
$\widehat{\overline{\Delta}}/\overline{\Delta}=2$. The model (19) is employed
in the stabilized mixed model [6, 7] with the modeled transport equation of
the SGS kinetic energy $\tau^{\mathrm{sgs}}_{\ell\ell}/2$. The second model
(20) is the Clark term, which is the leading term in the Taylor expansion of
the sum of Leonard and cross terms [46]; that is
$\overline{\overline{u}_{i}\overline{u}_{j}}-\overline{u}_{i}\overline{u}_{j}+\overline{\overline{u}_{i}(u_{j}-\overline{u}_{j})}+\overline{(u_{i}-\overline{u}_{i})\overline{u}_{j}}\simeq\sum_{\alpha}\overline{\Delta}_{\alpha}^{2}(\partial\overline{u}_{i}/\partial
x_{\alpha})(\partial\overline{u}_{j}/\partial
x_{\alpha})/12+O(\overline{\Delta}^{4})$. Inagaki and Kobayashi [8] examined
these two models in an a posteriori test of turbulent channel flows. They
found that the first model provides a better result than the second because
small-scale velocity fluctuation is significantly enhanced in the first one.

In addition to these two models, we also examine the following quadratic
nonlinear model based on the velocity gradient according to the explicit
algebraic SGS stress model [4, 5]:

$\displaystyle\tau^{\mathrm{ani}}_{ij}=-\tau^{\mathrm{sgs}}_{\ell\ell}\frac{\overline{s}_{im}\overline{w}_{mj}+\overline{s}_{jm}\overline{w}_{mi}}{\overline{s}^{2}},\
\ \overline{w}_{ij}=\frac{1}{2}\left(\frac{\partial\overline{u}_{i}}{\partial
x_{j}}-\frac{\partial\overline{u}_{j}}{\partial x_{i}}\right).$ (21)

This model does not contribute to the energy transfer without any artificial
treatments because
$(\overline{s}_{im}\overline{w}_{mj}+\overline{s}_{jm}\overline{w}_{mi})\overline{s}_{ij}=0$.
We set the arbitrary numerical coefficient to unity.

We can examine other models, including other normalizations, such as the
linear Clark model where the coefficient
$\tau^{\mathrm{sgs}}_{\ell\ell}/\tau^{\mathrm{a}}_{mm}$ is replaced with a
constant or some other nondimensional function. In addition, to remove the
energy transfer by the anisotropic term, we may be able to construct
treatments other than $\nu^{\mathrm{a}}\overline{s}_{ij}$. In this study, we
restrict ourselves our examination to the existing model expressions.

It is worth examining the profiles of the anisotropic redistribution term when
employing the commonly used eddy viscosities in LES because the definition of
eddy viscosity given by Eq. (4) is an ideal one. Then, we also examine the
anisotropic redistribution term by using the eddy viscosity based on the DSM
approach [47, 48]; namely, we calculate the anisotropic stress (5) by using
the following eddy viscosity

$\displaystyle\nu^{\mathrm{sgs}}=C\overline{\Delta}^{2}\sqrt{2\overline{s}^{2}},\
\ C=\max\left(\frac{\langle L_{ij}M_{ij}\rangle_{\mathrm{S}}}{2\langle M_{\ell
m}M_{\ell m}\rangle_{\mathrm{S}}},0\right),$ (22)

instead of using Eq. (4). Here,
$L_{ij}=\widehat{\overline{u}_{i}\overline{u}_{j}}-\widehat{\overline{u}}_{i}\widehat{\overline{u}}_{j}$,
$M_{ij}=\widehat{\overline{\Delta}}^{2}\sqrt{2\widehat{\overline{s}}^{2}}\
\widehat{\overline{s}}_{ij}-\overline{\Delta}^{2}\widehat{\sqrt{2\overline{s}^{2}}\
\overline{s}_{ij}}$, and $\langle\cdot\rangle_{\mathrm{S}}$ represents the
averaging over the $x$-$z$ plane. We clip the negative eddy viscosity
similarly to that in a posteriori tests of DSM. In contrast to the anisotropic
stress based on Eq. (4), that based on Eq. (22) allows the energy transfer
between GS and SGS. When the anisotropic redistribution based on Eq. (22)
(hereafter referred to as AR-DSM) becomes negligibly small, we can infer that
the DSM predicts well the interaction between the GS velocity fluctuation and
SGS even in the a posteriori test. In other words, when the AR-DSM
significantly contributes to the budgets, we can infer that the anisotropic
stress is necessary even when we employ a conventional eddy-viscosity model in
a posteriori tests. Furthermore, when the contribution of AR-DSM is positive
in the budgets, it cannot be modeled in terms of the eddy-viscosity
assumption.

Figure 13: A priori test of the anisotropic redistribution term for (a)
streamwise (b) wall-normal, and (c) spanwise components for the coarse filter
case. We also plot the models without the $\nu^{\mathrm{a}}$-related term of
Eqs. (19) and (20) in dashed lines.

Figure 13 shows the a priori prediction of the normal components of the
anisotropic redistribution. We refer to the models provided in Eqs. (19),
(20), and (21) as SGS-Reynolds, Clark, and SW, respectively. For reference, we
also plot models (19) and (20) without the $\nu^{\mathrm{a}}$-related term
(namely,
$\tau^{\mathrm{ani}}_{ij}=\tau^{\mathrm{sgs}}_{\ell\ell}\tau^{\mathrm{a}}_{ij}|_{\mathrm{tl}}/\tau^{\mathrm{a}}_{mm}$)
where they are allowed to contribute to the energy transfer including backward
scatter. For the streamwise component, all three models predict a positive
contribution, although they fail to predict a negative contribution observed
in fDNS in the vicinity of the wall. The SGS-Reynolds model provides a profile
most similar to the fDNS, whereas the Clark model shows a different profile.
The models without the $\nu^{\mathrm{a}}$ term provide a strongly negative
profile that is far from that of the fDNS. The profile of AR-DSM is rather
similar to that of EV dissipation in Fig. 6(b). Namely, it is almost negative
except near a positive peak at $y^{+}\approx 10$. The value of AR-DSM at
$y^{+}\approx 40$ is about $-0.02$, which is the half of EV dissipation of
fDNS there (see Fig. 6(b)). Note that the anisotropic redistribution of fDNS
is negligibly small at the point, and thus the total dissipation rate of
$R^{\mathrm{GS}}_{xx}$ due to the SGS stress is twice that estimated by DSM in
the a priori test. In other words, the eddy viscosity based on the DSM is less
dissipative than the ideal one given by Eq. (4).

The wall-normal component shown in Fig. 13(b) is well predicted by all models
including those without the $\nu^{\mathrm{a}}$ term. The profile of AR-DSM
also agrees with that of fDNS based on Eq. (4) qualitatively. For the spanwise
component shown in Fig. 13(c), the SW model provides a different sign than the
fDNS does, which indicates that it attenuates the spanwise GS velocity
fluctuation. The SGS-Reynolds and Clark models succeed in predicting a
positive profile observed in the fDNS, although their intensity is small. In
contrast to the Clark model, the SGS-Reynolds model can provide a positive
contribution without the $\nu^{\mathrm{a}}$ term. The positive value of AR-DSM
is smaller than that of fDNS based on Eq. (4) because the eddy viscosity based
on the DSM is less dissipative than that given by Eq. (4) as mentioned above.
If we employ a sufficiently dissipative eddy-viscosity model, the positive
contribution of anisotropic redistribution becomes large because the sum of EV
dissipation and anisotropic redistribution terms is positive at $y^{+}\approx
20$ for the fDNS as shown in Fig. 8. Therefore, we can infer that regardless
of the choice of the eddy-viscosity model, the anisotropic stress has a
significant contribution to the budgets even when we employ a conventional
eddy-viscosity model in a posteriori tests of coarse grid LES.

In general, the models with the $\nu^{\mathrm{a}}$ term provide better results
than those without it. Although the $\nu^{\mathrm{a}}$ term is an artificial
term introduced to remove energy transfer, it can improve the prediction of
the statistics. However, we do not consider that the models with
$\nu^{\mathrm{a}}$ term are always superior to other models in predicting the
budget. Furthermore, in the a posteriori test, we must determine the closed
expression of $\nu^{\mathrm{sgs}}$ without using DNS data. The a posteriori
performance of the model depends on the combination of the models of
$\nu^{\mathrm{sgs}}$ and $\tau^{\mathrm{ani}}_{ij}$. The main conclusion of
this study is that the anisotropic stress that predicts a positive
contribution to the spanwise GS velocity fluctuation is key to improving SGS
models.

##  V Conclusions

We have investigated the budget equation for grid-scale (GS or resolved scale)
Reynolds stress in turbulent channel flows. In the analysis, we have
decomposed the subgrid-scale (SGS) stress into two parts: the isotropic eddy-
viscosity term, which governs energy transfer between GS and SGS, and the
anisotropic term, which is separated from the energy transfer. According to
this decomposition, the SGS dissipation is decomposed into eddy-viscosity
dissipation and anisotropic redistribution terms. To clearly observe the role
of the anisotropy of SGS stress, we have employed a coarse-size filter in
addition to a medium-size filter. The filter length is chosen such that the
conventional eddy-viscosity models can fairly predict the mean velocity
profile for the medium filter case, whereas it fails for the coarse filter
case (see Appendix B).

The contribution of anisotropic redistribution to the budget of GS turbulent
kinetic energy for the medium filter is negligible, whereas in the coarse
filter, it has a small but positive contribution. A similar effect has been
observed for the streamwise component of GS Reynolds stress. For the wall-
normal component of GS Reynolds stress budget, the anisotropic redistribution
is negative in the entire region for both filter sizes. In contrast, for the
spanwise component of GS Reynolds stress budget, the anisotropic
redistribution is always positive for both filter sizes. Furthermore, the SGS
dissipation, which is the sum of eddy-viscosity dissipation and anisotropic
redistribution, is also positive in the near-wall region. For the coarse
filter case, the contribution of anisotropic redistribution is comparable to
that of pressure redistribution. Therefore, anisotropic SGS stress is
indispensable for reproducing the generation of GS spanwise velocity
fluctuation for LES using a coarse grid resolution.

It has been suggested that the positive contribution of SGS dissipation to the
budget is related to the coherent structures in the near-wall region of
turbulent shear flows [25, 37, 26, 38]. To determine the relationship between
coherent structures and positive contribution of anisotropic redistribution to
the budget, we have performed a spectral analysis of the GS Reynolds stress
budget. For the coarse filter case, the anisotropic redistribution is positive
in a relatively small-scale region in the near-wall region of the budget for
the streamwise component of GS Reynolds stress spectrum. This trend is
consistent with the streak breakdown process in the self-sustaining process of
wall-bounded turbulent shear flows [41, 42]. In the budget for spanwise GS
Reynolds stress spectrum in the near-wall region, the anisotropic
redistribution has a positive contribution where the streamwise length scale
is large and the spanwise length scale is close to the typical spacing of
streaks. The spanwise velocity fluctuation accompanied by a nonzero spanwise
wavenumber is related to streamwise vortices. Therefore, we can infer that the
positive contribution of anisotropic redistribution to the spanwise GS
Reynolds stress is related to the generation of streamwise vortices in the
self-sustaining process [41, 42]. Thus, this study suggests that the
anisotropic part of SGS stress is responsible for the generation of coherent
structures in wall-bounded turbulent shear flows.

We have performed an a priori test of several existing models of anisotropic
stress in terms of anisotropic redistribution. Among them, the model
expression employed in the stabilized mixed model [6] seems to be the best.
However, the intensity of the spanwise component is small. In addition, the
artificially introduced part that removes the energy transfer due to the
scale-similarity model contributes to the improvement of the profile of
anisotropic redistribution. A quadratic nonlinear model based on the velocity
gradient [4, 5] cannot reproduce the positive contribution to spanwise GS
Reynolds stress. In the future, we intend to develop a model that predicts all
the components of anisotropic redistribution. In conclusion, this study has
suggested that anisotropic SGS stress reproducing a positive contribution to
the GS Reynolds stress budget is key to improving SGS models. This viewpoint
can be a novel guiding principle in SGS modeling, particularly for coarse grid
cases.

In this study, we have not considered the filter in the wall-normal direction.
When we apply the filter in the wall-normal direction, the velocity gradient
in the vicinity of the wall is smoothed. Therefore, the interaction between
the mean and turbulent fields should be changed. To discuss the physics of
coarse LES based on the DNS in more detail, we have to consider wall-normal
filtering. However, to realize the filter that coincides with the actual LES,
we have to apply the inhomogeneous filter where the filter length changes
against the distance from the wall. This inhomogeneous filter is not
commutative with a differential operation and induces several additional terms
which are different from the stress term in the filtered continuity and
Navier-Stokes equations [14]. We have to assess the effect of such terms
arising from the commutation error. Furthermore, in wall-bounded turbulent
flows, the interaction between near-wall small eddies and the bulk flow is
also important [33]. The physics of wall-normal filtered velocity fields and
its subgrid-scale modeling should be discussed in a future study.

###### Acknowledgements.

K.I. was supported by a Grant-in-Aid for JSPS Fellows Grant No. JP21J00580.
H.K. was supported by Keio Gijuku Academic Development Funds.

##  Appendix A Decomposition of SGS diffusion term

Similar to the SGS dissipation term, we can decompose the SGS diffusion term
(7h) as follows:

$\displaystyle D^{\mathrm{SGS}}_{ij}$
$\displaystyle=D^{\mathrm{EV}}_{ij}+D^{\mathrm{AS}}_{ij},$ (A1)

where

$\displaystyle D^{\mathrm{EV}}_{ij}$ $\displaystyle=2\frac{\partial}{\partial
x_{\ell}}\left<\nu^{\mathrm{sgs}}\left(\overline{s}_{i\ell}\overline{u}_{j}^{\prime}+\overline{s}_{j\ell}\overline{u}_{i}^{\prime}\right)\right>,$
(A2a) $\displaystyle D^{\mathrm{AS}}_{ij}$
$\displaystyle=-\frac{\partial}{\partial
x_{\ell}}\left<\tau^{\mathrm{ani}}_{i\ell}\overline{u}_{j}^{\prime}+\tau^{\mathrm{ani}}_{j\ell}\overline{u}_{i}^{\prime}\right>.$
(A2b)

These two terms are referred to as eddy-viscosity and anisotropic stress
diffusions, respectively.

Figure 14: Decomposition of SGS diffusion term for (a) streamwise (b) wall-
normal, and (c) spanwise components for the coarse filter case.

Figure 14 shows the profiles of decomposed SGS diffusion terms. For the
streamwise component, the eddy-viscosity diffusion (EV) provides a profile
similar to SGS diffusion itself. However, the anisotropic stress diffusion
(AS) also has a value comparable to that of the eddy-viscosity term and
slightly alters the profile of SGS diffusion. The wall-normal component of SGS
diffusion is determined by anisotropic stress diffusion, whereas the eddy-
viscosity diffusion is negligible. For the spanwise component, the anisotropic
stress diffusion provides a profile similar to that of SGS diffusion. The
eddy-viscosity diffusion has the opposite sign to that of SGS and anisotropic
stress diffusions. However, for the spanwise component, the contribution of
SGS diffusion to the budget is negligible when compared with the other terms
as shown in Fig. 8. In contrast, the SGS diffusion contributes significantly
to the budget in the near-wall region for the wall-normal component as shown
in Fig. 7. Therefore, anisotropic SGS stress is also important to predict SGS
diffusion, particularly for the wall-normal component.

##  Appendix B Validation of DNS

Figure 15: Comparison between the present DNS and exiting database: profiles
of (a) mean velocity and (b) Reynolds stress.

Figure 15 shows the comparison between the present DNS and existing ones
performed by Moser et al. [49] and Abe et al. [50]. The database of the
nearest Reynolds number is chosen. Note that Abe et al. [50] used the fourth-
order finite difference in the $x$ and $z$ directions and the second-order
finite difference in the $y$ direction, which are the same condition as our
code, whereas Moser et al. [49] used the spectral discretization. The mean
velocity profiles are almost consistent for all codes. For the Reynolds
stress, the present DNS slightly overestimates the streamwise component in
$50<y^{+}<250$. However, the near-wall region seems to be well resolved.
Therefore, we infer that we can analyze the physics of this flow by our
present simulation, although the resolution is slightly coarser than that of
references.

##  Appendix C LES results of reference filter sizes

To observe the resolution dependence of eddy-viscosity models in the a
posteriori test, we performed LES using the dynamic Smagorinsky model (DSM)
[47, 48]. The numerical method and Reynolds number are the same as the DNS
provided in Sec. III.1. We chose a grid resolution corresponding to the
selected filter lengths. Namely, we performed the medium $(\Delta x^{+},\Delta
z^{+})=(52.4,26.2)$ and coarse $(\Delta x^{+},\Delta z^{+})=(105,52.4)$ cases.
We additionally performed a fine case with $(\Delta x^{+},\Delta
z^{+})=(26.2,13.1)$ to observe the detailed dependence on resolution. The grid
number in the $y$ direction is fixed at $N_{y}=96$ for all the cases. The
near-wall region is well resolved as $\Delta y^{+}<1$ for the first grid from
the wall, although the resolution is slightly coarser than that of DNS. The
test filter operation used to evaluate the eddy viscosity in the dynamic model
is calculated by retaining the first-order in the Taylor expansion (see Ref.
[8]) and is applied in all three directions. The filter length for the test
filter is set to twice the grid width in each direction.

Table 1: Numerical parameters for LES of DSM. Case | $N_{x}\times N_{y}\times N_{z}$ | $\Delta x^{+}$ | $\Delta y^{+}$ | $\Delta z^{+}$
---|---|---|---|---
fine | $96\times 96\times 96$ | 26.2 | 0.7–20 | 13.1
medium | $48\times 96\times 48$ | 52.4 | 0.7–20 | 26.2
coarse | $24\times 96\times 24$ | 105 | 0.7–20 | 52.4
Figure 16: Mean velocity profile for the LES of DSM with various grid
resolutions in comparison with that for DNS.

Figure 16 shows the mean velocity profile for the LES of DSM in comparison
with that for DNS. Both the fine and medium grid cases predict the mean
velocity of the DNS efficiently, whereas the coarse grid case overestimates
it. Thus, we can conclude that the medium grid resolution is sufficient for
predicting the mean velocity profile in turbulent channel flows.

Figure 17 shows the profiles of GS Reynolds stress for the LES of DSM in
comparison with those for DNS and filtered DNS (fDNS). The fine grid case
predicts all the nonzero components of GS Reynolds stress efficiently,
although the wall-normal and spanwise components are slightly underestimated.
In contrast, the medium grid case overestimates the streamwise component and
underestimates the wall-normal and spanwise components. These trends are often
observed in the LES of turbulent channel flows and are emphasized in the
coarse grid case. The analysis described in Secs III and IV suggests that this
issue can be resolved by employing an appropriate anisotropic SGS stress. The
medium grid case also overestimates the shear component. However, the mean
velocity is comparable to that of the DNS owing to the relatively small SGS
stress (the figure is not shown).

Figure 17: Profiles of GS Reynolds stress for the LES of DSM in comparison
with those for DNS and filtered DNS (fDNS) for (a) streamwise, (b) wall-
normal, (c) spanwise, and (d) shear components.

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