12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

INVESTIGATION OF SUBGRID-SCALE TURBULENT KINETIC ENERGYIN CHANNEL FLOWS

Kazuhiro Inagaki
Research and Education Center for Natural Sciences
Keio University
4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japanemail: inagakik@keio.jp

Hiromichi Kobayashi
Department of Physics & Research and Education Center for Natural Sciences, Hiyoshi Campus
Keio University
4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japanemail: hkobayas@keio.jp

ABSTRACTtion and dissipation terms in the SGS energy transport yields
the Smagorinsky model (Smagorinsky, 1963). Therefore, we
can interpret that the OEM incorporates the nonequilibrium
effect on the transport coefﬁcients. One may consider thatthe nonequilibrium effect resulting from the OEM signiﬁcantly
contributes to the robustness of the SMM against the grid resolution. However, the physical nature of OEMs has not been
investigated in detail yet. In general, solving an additionaltransport equation can be a numerical burden and lose the convenience of the model. Furthermore, the modeled transport
equation does not guarantee the positive semi-deﬁniteness of
the SGS energy. Consequently, the negative SGS energy must
be clipped when the eddy viscosity is modeled in terms of its
square root. If the OEM is reduced to an algebraic or zeroequation model of SGS energy in a physically reliable manner,
the convenience of the SGS model signiﬁcantly increases.
In the present study, we investigate the physical propertyof the OEM via the SMM with the ﬁltered DNS data in turbulent channel ﬂows. Thanks to the robustness of the SMM
against the grid resolution, we can investigate the SGS energy
in a coarse grid resolution in which the SGS energy is healthierthan the conventional LES with the eddy-viscosity models. We
also demonstrate the reduction of the SMM into the ZE-SMM
that excludes the transport equation for the SGS energy.

We investigate the subgrid-scale (SGS) turbulent kinetic
energy (hereafter referred to as SGS energy) and its transport
equation by employing the stabilized mixed model (SMM)
developed by Abe (2013). The prediction of the budget forthe modeled transport equation is qualitatively well comparedwith the ﬁltered DNS via the Gaussian ﬁlter. In addition, theproduction term balances with the dissipation rate in the outerregion, which suggests that the transport equation can be reduced by assuming the equilibrium between them. We investigate ﬁve algebraic expressions for the SGS energy. Amongthem, the equilibrium form with a newly introduced damping
function yields the best compared with the SMM. The zeroequation SMM (ZE-SMM) that employs the equilibrium form
instead of the transport equation for the SGS energy provides
quantitatively equivalent results as the original SMM. This
study paves the way for constructing a convenient anisotropic
SGS model that is robust even in coarse grid resolutions.

ANISOTROPIC STRESS MODELING AND SGS
ENERGY TRANSPORTIn the recent development of SGS modeling, several
studies demonstrated that the anisotropic stress term signiﬁcantly improves the grid sensitivity of large-eddy simulation(LES) (Abe, 2013; Inagaki & Abe, 2017; Marstorp et al.,2009; Montecchia et al., 2017). Abe (2019) and Inagaki &Kobayashi (2020) revealed that the anisotropic stress term essentially contributes to the generation of the grid-scale (GS)or resolved-scale Reynolds shear stress and energy spectrumat the high-wavenumber region. Therefore, the investigationof the anisotropic SGS stress models paves the way for thefurther development of LES of a more high-Reynolds number
turbulent ﬂows.
In these anisotropic models, the SGS energy is employedin the transport coefﬁcients of the modeled stress. Especiallyin the SMM (Abe, 2013), the SGS energy is obtained bynumerically solving its transport equation, which is referred
to as the one-equation model (OEM). In the OEM with theeddy-viscosity assumption, the balance between the produc-

Introduction of SMM and OEMsgs

In the SMM, the SGS stress τi j (= ui u j − ui u j ) is modeled as follows:τ a | + 2ν a si j
2
sgs
eat
sgs i j tl
τi j = ksgs δi j − 2ν sgs si j + τieat
τ
,
,
=
2k
j
ij
a
3
τℓℓ
νa = −

τiaj |tl si j
2sℓm sℓm

sgs, τiaj = (ui − bui )(u j − bu j ).

(1)

Here, ksgs (= τℓℓ /2) and ν sgs denote the SGS energy andeddy viscosity, respectively. · and ˆ· denote the ﬁlter and test
ﬁlter operations, respectively. ui is the velocity ﬁeld andsi j [= (∂ ui /∂ x j + ∂ u j /∂ xi )/2] is the GS strain rate. The SGS

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Numerical parameters. The domain size is Lx ×Ly ×
Lz = 2π h × 2h × π h. LR and MR denote the low and medium
resolutions, respectively.

eddy viscosity ν sgs is expressed by√
ν sgs = Csgs fν ∆ ksgs , fν = 1 − exp[−(dε /A0 )2/(1+C0 ) ], uε y y C0dε =, uε = (νε sgs )1/4 .
(2)ν∆Here, ∆ denotes the ﬁlter length scale and ν is the kinematic
viscosity. ε sgs will be provided later. Inagaki & Abe (2017)
suggested the following parameters: ∆ = (∆x∆y∆z)1/3 , Csgs =0.075, A0 = 13, and C0 = 1/3. ∆x, ∆y, and ∆z are the grid
spacing in each direction in a Cartesian coordinate. The ﬁlter
length scale for the test ﬁlter is set to be ∆ˆ = 2∆.
In OEMs including the SMM, ksgs is obtained by numerically solving the following transport equation:∂ ksgs∂sgs=−(ui ksgs ) − τi j si j − ε sgs
∂t
∂ xi
∂
∂ 2 ksgs
t,sgs
p,sgs
(T j + T j
)+ν
,
−
∂xj
∂ xi ∂ xi

Case

Reτ

Nx × Ny × Nz

∆x+

∆z+

SMM LR400

24 × 64 × 16

105

79

SMM MR400

48 × 64 × 32

52

39

EVM LR400

24 × 64 × 16

105

79

EVM MR400

48 × 64 × 32

52

39

DNS

400

256 × 192 × 256

9.8

4.9

SMM1000

1000

48 × 96 × 32

131

98

SMM2000

2000

96 × 128 × 64

131

98

25
DNS
SMM LRSMM MR
EVM LR
EVM MR

(3)
20

t,sgs
where ε sgs = ν [(∂ ui /∂ x j )2 − (∂ ui /∂ x j )2 ], T j= u j ui ui /2 −
sgs
p,sgs
= pu j − p u j . We refer to theu j ui ui /2 − ui τi j , and T j

U+
x

15

terms on the right-hand side of Eq. (3) as convection, production, dissipation, turbulent diffusion, pressure diffusion, andt,sgs
viscous diffusion, respectively. In OEMs, ε sgs and T j
+
p,sgs
Tj
are modeled as follows:(ksgs )3/2 2ν ksgs
+
,
y2
∆
√
∂ ksgs
t,sgs
p,sgs
.
Tj + Tj= −Ck fν ∆ ksgs
∂xj

ε sgs = Cε

10

5

0

(4)

1

10 +

100

y

(5)

Figure 1. Mean velocity proﬁles for the DNS, SMM, and
EVM at Reτ = 400 in each grid resolution.

Here, y denotes the distance from the solid wall. The second
term on the right-hand side of Eq. (4) is the wall correction
term. We set Cε = 0.835 and Ck = 0.1 (Inagaki & Abe, 2017).
Note that in the SMM, τieat
j si j = 0 owing to the stabilization
treatment of the ν a -related term. Hence, the production always
sgs
yields a positive. Namely, −τi j si j = 2ν sgs s2 ≥ 0 where s2 =si j si j .

ﬁltered DNS, we employ the Gaussian ﬁlter only in the x andz directions where the cut-off wavenumber corresponds to the
maximum wavenumber of the LES.

Numerical results
Figure 1 shows the mean velocity proﬁles for the DNS,
SMM, and EVM at Reτ = 400. The EVM overestimates themean velocity in the LR. In contrast, for the SMM, the mean
velocity proﬁles are comparable to the DNS for both resolutions. Figure 2 shows the budgets for the SGS energy transport
equation for the ﬁltered DNS and SMM in the LR. Although
the SMM overestimates the production and dissipation rates in
the near-wall region, the prediction of the SMM is fairly wellcompared with the ﬁltered DNS. Furthermore, the production
term balances with the dissipation rate in y+ > 50, which suggests that the transport equation can be reduced by assuming
the production–dissipation equilibrium.
Figure 3 shows the correlation between the production
and dissipation terms in the SGS energy transport equation,
where pthe correlation coefﬁcient is deﬁned by cor( f , g) =⟨ f ′ g′ ⟩/ ⟨ f ′2 ⟩⟨g′2 ⟩ with f ′ = f − ⟨ f ⟩. For the ﬁltered DNS,
the correlation is low. Hence, the local equilibrium assumption
is poor, as it is classically discussed. This result is obvious because the backscatters often occur in the ﬁltered DNS, whichmust not balance with the local dissipation. In contrast, theLESs of both the SMM and EVM provide a high correlation

NUMERICAL SIMULATION
We perform LESs of the SMM and DNS in turbulent
channel ﬂows in a Cartesian coordinate with a staggered grid
system. x, y, and z denote the streamwise, wall-normal, andspanwise directions, respectively. For details of the numerical
schemes, please refer to Inagaki & Kobayashi (2020, 2022).
For the reference LES of the eddy-viscosity model, we also
perform the case with τieat
j = 0 and Csgs = 0.042 in the SMM,
which is referred to as the EVM. Note that Csgs = 0.042 is thebest value for the eddy-viscosity-based OEMs (Inagaki, 2011;
Inagaki & Kobayashi, 2020). The Reynolds
p number is set to
be Reτ (= uτ h/ν ) = 400 where h, uτ (= |∂ Ux /∂ y|wall |), andUx (= ⟨ux ⟩) denote the channel half width, wall-friction velocity, and mean velocity, respectively. For the LES at higher
Reynolds numbers, we also perform the cases at Reτ = 1000
and 2000. The statistical average ⟨·⟩ is taken over the x–z plane
and time. Details of the parameters are provided in Table 1.
Here and hereafter, the values with the superscript “+” indicate those normalized by ν and uτ . For the analysis of the2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.2

SMM LRSMM MR
SMM1000
SMM2000
new damping cs=7.3
new damping cs=7.6

1.4〈ksgs〉/〈(2Csgs/Cε)∆2s-2〉

〉 /∂t

0.05

sgs +

∂〈k

+

0.15
0.1

1.6

production
dissipation
convection
turbulent + pressure diffusions
viscous diffusion
dissipation w/o wall term

0

1.2
1
0.8
0.6
0.4
0.2

-0.05

0

50

100

150

200
+

y

250

300

350

0

400

1

s10

〈y 〉

100

Figure 4. Ration of the SGS energy to the local equilibrium
expression for the SMM with respect to the mean distancefrom the wall normalized by the GS Kolmogorov length scale⟨ys ⟩.

Figure 2. Budgets for the SGS energy transport equation for
the DNS (solid lines with symbols) and SMM (dashed lines
with symbols) at Reτ = 400 in the LR. The dash-dotted line
represents the dissipation term without wall correction in the
SMM; ⟨Cε (ksgs )3/2 /∆⟩.

tion points:
1

fk =1 − exp[−(ys /as )2 ] s y(s2 )1/4
,y =.
1 + exp[−bs ys + cs ]
ν 1/2

(6)

cor(P

,ε

sgs sgs)

0.8

Here, ys can be interpreted as the distance from the wall normalized by the GS Kolmogorov length scale. Namely, ys =y/η gs with η gs = (νε gs /2)1/4 and ε gs = 2ν s2 . The numeratorof fk realizes the near-wall asymptote fk ∼ O(y2 ), whereas thedenominator represents the strong damping of the SGS energy
in the vicinity of the wall.
Figure 4 shows the proﬁles of the ratio of the SGS energy
to the local equilibrium expression for the SMM with respect
to the mean distance from the wall normalized by the GS Kolmogorov length scale ⟨ys ⟩. This ratio corresponds to the exact
damping behavior in the near-wall region. Note that y+ = 50
and 100 correspond to ⟨ys ⟩ = 10 and 20, respectively. Regardless of the Reynolds number, the ratio almost collapses to asingle curve in the near-wall region. The black dashed linewith crosses depicts the best ﬁt curve obtained by minimizing
the mean square error of the SGS energy between the exact
proﬁle and model expression for the SMM at Reτ = 400 in the
LR. The parameters yield as = 0.6, bs = 0.77, and cs = 7.3.
However, in the a posteriori test that employs the algebraic
expression of SGS energy with this damping function, themean velocity is slightly underestimated with these parameters. Therefore, in the a posteriori test, we reﬁne the parameteras cs = 7.6. The damping function with the reﬁned parameteris plotted in Fig. 4 in the gray solid line with pluses.
In LES, we can employ the scale-similarity models (Bardina et al., 1983) as a representative quantity to express themodel of SGS energy. The dynamic procedure (Marstorp
et al., 2009; Montecchia et al., 2017) or invariants of the velocity gradient (Kobayashi, 2005; Silvis et al., 2017) are also
useful in expressing the near-wall asymptote of the SGS energy. Here, we compare the following algebraic expressions
for the SGS energy calculated in the SMM with the result via
the transport equation:

0.6

0.4

0.2

0

0

filtered DNS LR
filtered DNS MR
SMM LRSMM MR
0.2
0.4

y/h

0.6

EVM LR
EVM MR
SMM1000
SMM2000
0.8

1

Figure 3. Correlation coefﬁcient between the production Psgs
and dissipation ε sgs for the ﬁltered DNS, SMM and EVM.
Note that for the LESs, the dissipation term excludes the wallcorrection term; namely, ε sgs = Cε (ksgs )3/2 /∆ is employed.
that yields 80%. In addition, the correlation is high at highReynolds-number cases. This result partly emanates from themodeling of the production term given by 2ν sgs s2 , which isalways positive and excludes the backscatter. Therefore, if theOEMs employ the eddy-viscosity-based production term, thelocal production–dissipation equilibrium assumption can be agood model.

Reduction of SMM employing an algebraic expression for SGS energy
In the region away from the wall, the local production–
dissipation equilibrium in the SGS energy transport equation
2
yields ksgs = (2Csgs /Cε )∆ s2 . However, this model cannot
be applied in the near-wall region because s2 does not decrease, whereas ksgs must decrease as O(y2 ). A primitive
approach to representing the proper near-wall behavior is toemploy a damping function. A basic damping function is
1 − exp[−(y+ /A)2 ] with a constant A. However, the wallfriction velocity uτ is not always a suitable variable for thedamping function because it yields zero at a separation point(see e.g. Inagaki, 2011). Here, we introduce a new damping
function that is applicable to the turbulent ﬂows with separa-

1. SGS-Reynolds formui )(ui − bui ),CS (ui − b(7)

2. dynamic form (Marstorp et al., 2009; Montecchia et al.,
2017)

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) 2.5

35
filtered DNS
SMM
SGS-Reynolds
dynamic
coherent structure
vortex stretchingequilibrium (a priori)
equilibrium (a posteriori)〈ksgs〉+

1.5

30
25
20

U+
x

2

15

1

10
0.5
5
0

0

50

100

150

(b)1.8

250

y

300

350

0

400

filtered DNS
SMM
SGS-Reynolds
dynamic
coherent structure
vortex stretchingequilibrium (a priori)
equilibrium (a posteriori)

1.6
1.4
1.2

〈ksgs〉+

200
+

1

0.4
0.2
50

100

150

200
+

250

y

300

350

400

Figure 5. Proﬁles of the SGS energy for the ﬁltered DNS,
SMM, and several model expressions calculated in the SMM
in (a) LR and (b) MR at Reτ = 400.

2

2CD ∆ s2 , CD =1 ⟨LM⟩,
4 ⟨M 2 ⟩2 2
2uibui , M = ∆ˆ ŝ − ∆ sb2 ,L = udi ui − b(8)

3. coherent structure form (Kobayashi, 2005)
2

CCS FCS ∆ s2 , FCS =|w2 − s2 |,w2 + s22(si j ω j )2s2 ω 2ℓPerformance of ZE-SMM,

Figure 6 shows the mean velocity proﬁles for the DNS,
SMM, and ZE-SMM. For all cases, the ZE-SMM provides
a comparable proﬁle compared with the original OEM-based
SMM and well predicts the proﬁle of the DNS. The total turbulent kinetic energy K is deﬁned as(10)

5. equilibrium form with the new damping function
fk

2Csgs 2 2
∆ s ,
Cε

1000

(9)

4. vortex stretching form (Silvis et al., 2017)
CVS FVS ∆ s2 , FVS =y+with the SMM does not necessarily predict the proﬁle of the
SGS energy for the ﬁltered DNS. Although the SGS-Reynolds
form seems to be reasonable in the LR, it excessively overestimates the proﬁle. Hence, if we employ the SGS-Reynolds
form instead of the OEM, the model will highly depend on thegrid resolution. The dynamic, coherent structure, and vortex
stretching forms fail to reproduce the sharp proﬁle in the nearwall region. In contrast, the equilibrium form succeeds in predicting the sharp proﬁles like the ﬁltered DNS and SMM. For
the equilibrium form, we also plot the result of the a posteriori
test, which is almost similar to that of the a priori test. Notethat the a priori test indicates the prediction using the data of
the SMM, whereas the a posteriori test indicates the result of
the simulation of the ZE-SMM that employs the equilibrium
model for the SGS energy instead of solving its transport equation. Among ﬁve model expressions (7)–(11), the equilibrium
form with the new damping function yields the best comparedwith the SMM. Therefore, we expect that the equilibrium form
with the new damping function (11) can provide a similar result compared with the original OEM-based SMM.

0.6

0

10

Figure 6. Mean velocity proﬁles for the DNS, SMM, and ZESMM. The results for Reτ = 1000 and 2000 are shifted by afactor of ﬁve and ten, respectively. The DNSs at Reτ = 1000
and 2000 are performed by Lee & Moser (2015).

0.8

0

1

DNS
SMM LR
ZE-SMM LR
SMM MR
ZE-SMM MR
SMM1000
ZE-SMM1000
SMM2000
ZE-SMM2000
100

(11)

K=

where wi j = (∂ ui /∂ x j − ∂ u j /∂ xi )/2, ω i = εi jℓ ∂ uℓ /∂ x j , andfk is provided by Eq. (6) with as = 0.6, bs = 0.77, and cs = 7.6.
All model expressions satisfy the near-wall asymptote ksgs ∼O(y2 ). The model constants are set to be CS = 2.5, CCS = 0.7,
and CVS = 1.1.
Figures 5(a) and (b) show the proﬁles of the SGS energy
for the ﬁltered DNS, SMM, and several model expressions calculated in the SMM provided by Eqs. (7)–(11) in the LR andMR, respectively, at Reτ = 400. In the LR, the proﬁle of the
SMM is reasonable compared with the ﬁltered DNS. However,in the MR, the SMM overestimates the SGS energy comparedwith the ﬁltered DNS. Hence, we can observe that the OEM

1 ′ ′u u + ⟨ksgs ⟩ .
2 i i

(12)

This decomposition is unique when the ﬁltering is employed
only in the directions that the spatial average is taken. In thepresent LESs, we assume that the right-hand side of Eq. (12)yields the total turbulent kinetic energy despite the employment of an implicit ﬁltering operation in the y direction. Figure 7 shows the proﬁles of the total turbulent kinetic energy for
the DNS, SMM, and ZE-SMM. The ZE-SMM provides almost
the same proﬁle as the original SMM for all cases, although the
SGS energy decreases as seen in Fig. 5. For high-Reynoldsnumber cases of Reτ = 1000 and 2000, both the SMM and ZESMM slightly underestimate the total turbulent kinetic energy
in the outer region. Further improvement of the SGS model

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
18
DNS
SMM LR
ZE-SMM LR
SMM MR
ZE-SMM MR

16
14

SGS energy and its transport under the condition that the SGS
energy is healthier than the conventional LESs. For the LR
case at Reτ = 400, the peak value of the SGS energy is approximately 50% of that of the total turbulent kinetic energy(see Figs. 5 and 7) even though the mean velocity proﬁle iscomparable to the DNS. The budget of the modeled transport
equation for the SGS energy is qualitatively comparable withthe exact one calculated in the ﬁltered DNS. In the budget, theproduction term balances with the dissipation rate in the outerregion. For the ﬁltered DNS, as is well known, the correlation
between the production and dissipation terms is low. In contrast, for the LESs employing the OEM, the production term
correlates well with the dissipation. This result indicates thatthe local production–dissipation equilibrium assumption canbe a good approximation for the OEMs, although this assumption does not hold in the ﬁltered DNS.
To construct an algebraic expression for the SGS energy
based on the local equilibrium assumption, we introduced a
new damping function where the distance from the solid wall
is normalized by the GS Kolmogorov length scale. The ratio of the SGS energy to the local equilibrium model almost
collapses to a single curve when it is plotted with respect tothe mean distance from the solid wall normalized by the GS
Kolmogorov length scale. Based on this curve, we developed
a new damping function. We compared the proﬁles of the
SGS energy for the ﬁltered DNS, SMM, and several model
expressions including the local equilibrium model with thedamping function. Among the model expressions provided in
the present analysis, the equilibrium model with the damping
function yields the best prediction compared with the result of
the transport equation in the SMM.
Finally, we performed the reduced model of the SMM
that employs the equilibrium form with the damping function for the SGS energy instead of solving the transport equation, which is referred to as the ZE-SMM. The ZE-SMM has
the equivalent performance compared with the original OEMbased SMM. Namely, the ZE-SMM provides quantitatively
similar results to the SMM for the proﬁles for the mean velocity, total turbulent kinetic energy, and streamwise component of the GS Reynolds stress spectrum. Consequently, thepresent study revealed that the conventional SGS energy transport equation is not essential in realizing the preferable properties of the SMM. Although the investigation has been conducted only in turbulent channel ﬂows, the present reduction
from the OEM to the equilibrium model is expected to be valid
in other turbulent ﬂows (for details, see Inagaki & Kobayashi,2022).
The success of the reduction using the local equilibrium
assumption partly results from the modeling of the production
term expressed in terms of the eddy viscosity and strain rate,
which does not provide the backscatter. There is the possibility that the OEM that allows the backscatter improves the SGS
models. However, in the OEMs, the SGS energy must be positive because its square root is usually employed in the eddy
viscosity. The modeling of the SGS energy transport equationthat allows both the backscatter and positive semi-deﬁnitenessis challenging. The local equilibrium expression is a primitivemodel for the SGS energy. The present study paves the way
for the further development of the SGS modeling that requires
the model expression of SGS energy.

SMM1000
ZE-SMM1000
SMM2000
ZE-SMM2000

12

K+

10
8
6
4
2
0

0

0.2

0.4

0.6

0.8

1

y/h
Figure 7. Proﬁles of the total turbulent kinetic energy for the
DNS, SMM, and ZE-SMM. The results for Reτ = 1000 and2000 are shifted by a factor of ﬁve and ten, respectively. The
DNSs at Reτ = 1000 and 2000 are perforsmed by Lee & Moser(2015).

is required to reproduce a more accurate proﬁle of the total
turbulent kinetic energy.
Inagaki & Kobayashi (2020) suggested that the enhancement of small-scale turbulence contributes to the robustness of
the SMM against the grid resolution. The GS Reynolds stress
spectrum is deﬁned by
E
D
′ ′∗
,
EiGS
j (kx , y) = ℜ ũi ũ j(13)where we consider the Fourier transformation only in the x
direction:ũi (kx , y, z) =

1
Lx

Z Lx
0

dx ui (x, y, z) exp[−ikx x].

(14)

We focus on the most energetic part, that is, the streamwise
GS . Figure 8 shows the streamwise component ofcomponent Exx
the premultiplied GS Reynolds stress spectrum for the ﬁltered
DNS, SMM, ZE-SMM, and EVM. Here, the λx (= 2π /kx ) is
the wavelength. The ﬁltered DNS provides a large intensity ofthe spectrum even in the small scales λx+ < 300. The result of
the ZE-SMM is almost the same as that of the SMM, whichindicates that the reduction of the OEM by employing the algebraic model (11) does not change the statistical property of
the SMM. Both the SMM and ZE-SMM also provide a large
intensity in the small scales λx+ < 300 in the near-wall regiony+ < 50. In the outer region, the SMM and ZE-SMM provide asmaller intensity than the ﬁltered DNS. To predict the large intensity in the outer region, we have to further improve the SGS
model. For the EVM, the spectrum rapidly decreases in thesmall scale λx+ < 400. This underestimation of the spectrum
results from the poor physical role of the eddy-viscosity models (see Inagaki & Kobayashi, 2020). The anisotropic stress isessential in the reproduction of small-scale turbulence.

CONCLUSIONS
We investigated the transport equation for the SGS energy
in turbulent channel ﬂows via the DNS and LES employing theSMM. Thanks to the robustness of the SMM against the grid
resolution, we succeeded in examining the properties of the

ACKNOWLEDGMENTS
K.I. was supported by a Grant-in-Aid for JSPS Fellows
Grant No. JP21J00580.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(b)

(a)
1

100

1

100

0.1

y+

y+

0.1
10

10

0.01

0.01

1

1

0.001

λ+
x

10

0.001

3

λ+
x

10

3

(d)

(c)
1

100

1

100

0.1

y+

y+

0.1
10

10

0.01

0.01

1

0.001
+ 10

3

1

0.001
+ 10

λx

3

λx

GS for (a) ﬁltered DNS
Figure 8. Contour maps of the streamwise component of the premultiplied GS Reynolds stress spectrum kx Exx(b) SMM, (c) ZE-SMM, and (d) EVM in the LR at Reτ = 400.

REFERENCES

turbulent transition region. Int. J. Heat Fluid Flow 64, 137–
152.
Kobayashi, H. 2005 The subgrid-scale models based on coherent structures for rotating homogeneous turbulence andturbulent channel ﬂow. Phys. Fluids 17, 045104.
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of
turbulent channel ﬂow up to reτ ≈ 5200. J. Fluid Mech. 774,
395–415.
Marstorp, L., Brethouwer, G., Grundestam, O. & Johansson,
A. V. 2009 Explicit algebraic subgrid stress models with application to rotating channel ﬂow. J. Fluid Mech. 639, 403–
432.
Montecchia, M., Brethouwer, G., Johansson, A. V. & Wallin,
S. 2017 Taking large-eddy simulation of wall-bounded
ﬂows to higher Reynolds numbers by use of anisotropyresolving subgrid models. Phys. Rev. Fluids 2, 034601.
Silvis, M. H., Remmerswaal, R. A. & Verstappen, R. 2017
Physical consistency of subgrid-scale models for large-eddy
simulation of incompressible turbulent ﬂows. Phys. Fluids
29, 015105.
Smagorinsky, J. 1963 General circulation experiments with theprimitive equations. I. The basic experiment. Mon. Weather
Rev. 91, 99–164.

Abe, K. 2013 An improved anisotropy-resolving subgrid-scalemodel with the aid of a scale-similarity modeling concept.
Int. J. Heat Fluid Flow 39, 42–52.
Abe, K. 2019 Notable effect of the subgrid-scale stressanisotropy on mean-velocity prediction through budgetof the grid-scale Reynolds-shear stress. Phys. Fluids 31,
105103.
Bardina, J., Ferziger, J. H. & Reynolds, W. C. 1983 Improved
turbulence models based on large eddy simulation of homogenous, incompressible, turbulent ﬂows. Report TF-19.
Thermosciences Division, Dep. of Mech. Eng., Stanford
University, Stanford, California .
Inagaki, K. & Kobayashi, H. 2020 Role of various scalesimilarity models in stabilized mixed subgrid-scale model.
Phys. Fluids 32, 075108.
Inagaki, K. & Kobayashi, H. 2022 Transport and modeling ofsubgrid-scale turbulent kinetic energy in channel ﬂows. AIPAdvances 12, 045222.
Inagaki, M. 2011 A new wall-damping function for large eddy
simulation employing Kolmogorov velocity scale. Int. J.
Heat Fluid Flow 32, 26–40.
Inagaki, M. & Abe, K. 2017 An improved anisotropyresolving subgrid-scale model for ﬂows in laminar–

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