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

WALL-HEATED/COOLED EFFECTS ON LOW-FREQUENCY PRESSURE
FLUCTUATIONS IN SHOCK-WAVE AND TURBULENT BOUNDARY LAYERINTERACTIONS

Ryo Hirai
Department of Aerospace Engineering
Tohoku University
Sendai Miyagi 980-8579, Japanryo.hirai.q7@dc.tohoku.ac.jp

Soshi Kawai
Department of Aerospace Engineering
Tohoku University
Sendai Miyagi 980-8579, Japankawai@tohoku.ac.jp

ABSTRACT2014). Regarding the upstream forcing mechanism, some previous studies have shown correlations between upstream flow
properties and the shock-foot motion (Brusniak and Dolling,
2006; Touber and Sandham, 2011). Also, the existence oflong coherent flow structures, whose length is approximately
40 times as long as boundary layer thickness, is observed experimentally at attached boundary layer flows (Ganapathisubramani et al., 2007 & 2009). The authors reported that thelarge structures flow into the shock interacted regions and influence the upstream or downstream shifts of the separation
point. In contrast, some downstream forcing mechanisms focus on boundary layer separations that lift wall turbulence toouter boundary layers and generate separation bubbles. For example, a low-frequency oscillation model has been proposed
based on the properties of fluid entrainment in shock-induced
separation bubbles (Piponniau et al., 2009). Another work proposed an acoustic feedback mechanism associated with incident shock oscillations by passages of uplifted vortex structures (Pirozzoli & Grasso, 2006).

Direct numerical simulations of shock-wave and turbulent
boundary layer interactions with wall heat flux are performed
to investigate wall heat flux effects on low-frequency dynamics
of shock interacted flows. We compare three wall temperaturecases of quasi-adiabatic, heated, and cooled conditions under
the freestream Mach number M∞ = 2.28 and Reynolds numberReθ ≈ 5, 000. The results focusing on low-frequency dynamics of flow and pressure fields suggest that boundary layer separations and reflected shock are associated with low-frequency
wall pressure fluctuations. Also, high-frequency wall pressure fluctuations originate from turbulent structures at separated shear layers and attached boundary layers. The results
and discussion suggest that wall temperature effects on pressure gradients of reflected shock and the semi-local Reynolds
number can change the low- and high-frequency wall pressure
fluctuations, respectively.

INTRODUCTION

As shown above, the low-frequency unsteadiness of
SWBLIs has been actively researched in the last two decades.
However, most of the studies address cases of adiabatic wallconditions. In a practical situation of rocket nozzles, a wallcooling system is often equipped to decrease the damage
caused by thermal loads. The wall heat flux can affects shockinteracted boundary layer flows at an inner wall of the nozzle
and changes the characteristics of low-frequency pressure fluctuations, thus, we need to understand the detailed contributionsof wall heat flux to the flow physics of SWBLIs. In a previous
study of SWBLIs with wall-thermal conditions, wall heating
and cooling effects on the shock-induced flow separations arereported; the separated regions get larger and smaller in heatedand cooled wall cases, respectively (Bernardini et al., 2016).
Also, Jaunet and Volpiani et al. reported influences of wallheat flux on wall pressure fluctuations whose spectral distributions shift to lower and higher frequencies in the heated andcooled case, respectively, compared with the adiabatic case(Jaunet et al., 2014; Volpiani et al., 2018). However, detailed

In many aerospace applications, shock-wave and turbulent boundary layer interactions (SWBLIs) significantly affectthe aerodynamic and thermodynamic performance by inducing
flow separations that increase drag and heat exchanges at wall
surfaces. Also, the unsteady flow fields provide pressure fluctuations, including low-frequency components at least twoorder lower than dominant frequencies of attached turbulent
boundary layers. The low-frequency pressure fluctuations canlead to structural fatigue when coupling to resonant frequencies of aircraft wings or overexpanded rocket nozzles (Deck,2009). Thus, detailed mechanisms of low-frequency unsteadiness in SWBLIs should be elucidated to reduce undesirable
structural oscillations.
Flow physics of SWBLIs has been investigated by experimental and numerical studies over the last few decades. Recently, driving mechanisms of the low-frequency unsteadinesshave been proposed in two categories: a forcing mechanism
from upstream or downstream (Clemens and Narayanaswamy,

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Transition location of wall temperature
Rescale and reintroduction

y/δ0

15

0

Shock impingement location xsh

Extract for inflow

Driver zone

M∞ = 2.28
Reθ,inlet ≈ 2500

Incident shock

Mean inlet boundary layer thickness ≈ δ00x/δ05052 55 0

8

Reflected shock
55

35

x/δ0

Quasi adiabatic/heated/cooled wall

Quasi adiabatic wall

Figure 1.

Test zone

32.7∘

Computational domain of present DNS. Mean streamwise velocity contours.

mechanisms of wall heat flux to affect flow physics driving the
low-frequency unsteadiness of SWBLIs have not been elucidated. Therefore, We should investigate how wall heating or
cooling affects the SWBLIs to evaluate wall heat flux contributions to controlling low-frequency oscillations induced byshock interactions.
In this study, direct numerical simulation (DNS) of
oblique shock-wave and turbulent boundary layer interactions
on a heated or cooled flat wall is conducted to investigate wallheat flux effects on low-frequency dynamics of shock-wave
interacted turbulent boundary layer flows. We perform threedifferent wall temperature cases (i.e., quasi-adiabatic, heated,and cooled) under the freestream Mach number M∞ = 2.28
and Reynolds number Reθ ≈ 5, 000 conditions and evaluate
differences in mean flow fields and low-frequency fluctuation
properties by comparing the cases.

qj = −

µ
=
µ∞

T
T∞

)3

2

T∞ + T1,
T + T1(7)Case settings
Figure 1 shows the schematic of the present simulationsof oblique shock-wave and turbulent boundary layer interaction on a flat plate with wall heat flux. There are two divided zones (i.e., driver and test zones) whose domain sizeis (Lx , Ly , Lz ) = (55δ0 , 15δ0 , 6δ0 ), where δ0 is approximately
equal to the inlet boundary layer thickness of driver zone. In
driver zone (on left in Fig. 1), the incoming turbulent boundary
layers are generated by the rescaling-reintroduction method(Urbin & Knight, 2001) under quasi-adiabatic wall conditions.
The rescale length x = 52δ0 is decided to eliminate the artificial streamwise contamination caused by the reintroduction(Simens et al., 2009). The turbulent boundary layer flows at
x = 50δ0 downstream of driver zone are given directly to theinlet of test zone (on right in Fig. 1). In test zone, the wall
temperature is transitioned from the quasi-adiabatic condition
to the heated and cooled conditions at x = 8δ0 downstream.
The quasi-adiabatic, heated, and cooled wall temperatures are
difined basd on the recovery temperature as

(1)
(2)
(3)

where ρ is the density, p is the pressure, and E is the total energy. The streamwise, wall-normal, and spanwise coodinates
are denoted by x j = (x, y, z) respectively, and ui = (u, v, w)
are the corresponding velocity components. The pressure p
is given by the equation of state of ideal gas as[]
M2
Tr = T∞ 1 + (γ − 1)r ∞ ,2(8)

where r (= 0.9) is the recovery coefficient. Also, incident
shock-wave generated at the upper boundary impinges on thelower wall at xsh = 35δ0 downstream with 32.7 angles whenassuming the inviscid wall conditions. In both zones, periodic
and outflow conditions are imposed in the spanwise and outlet
boundaries, respectively.
The sixth-order compact scheme is used for spatial discretization with the eighth-order compact low-pass filter,
where a filter coefficient is set to 0.495 (Lele, 1992; Gaitonde& Visbal, 1999; Kawai & Fujii, 2008). The localized artificial diffusivity method is employed to capture shock waves(4)

where R is the gas constant and T is the temperature. The
viscous stress tensor τi j and heat flux vector q j are given as[ (])
1 ∂ ui ∂ u j
1 ∂ ukτi j = 2µ+−δi j ,
2 ∂ x j ∂ xi
3 ∂ xk(where T∞ = 255.0 [K] and T1 = 110.4 [K]. The subscript ∞denotes the freestream quantities.

In the present study, we carry out direct numerical simulations by solving the following compressible Navier-Stokes
equations as

p = ρ RT,(6)

where µ is the dynamic viscosity, a is the speed of sound, andPr (= 0.72) is the Prandtl number. The dynamic viscosity isgiven by Sutherland’s law as

COMPUTATIONAL STRATEGY
Governing equations()∂ ρ ∂ ρu j
+
= 0,
∂t
∂xj
(
)
∂ (ρ ui ) ∂ ρ ui u j + pδi j − τi j
+
= 0,
∂t
∂xj
)
(
∂ (ρ E) ∂ ρ Eu j + pu j + q j − ui τi j
+
= 0,
∂t
∂xj

µ
∂ a2
,
Pr (γ − 1) ∂ x j(5)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Flow and grid parameters for present DNS. Lint , the interaction length; N, number of grid points in test zone; subscript w,
properties taken at wall.

Case

M∞

Reθ

Reτ

Tw /Tr

Lint /δ0

∆x+

∆y+
w

∆y+

∆z+

Nx × Ny × Nz

Quasi-adiabatic

2.28

5129

909

1.0

5.33

4.35

0.44

5.26

4.35

5501 × 388 × 611

Heated

2.28

5244

412

2.0

8.66

1.86

0.19

2.25

1.86

5501 × 388 × 611

Cooled

2.28

5066

2149

0.5

3.45

5.34

0.53

6.42

5.34

11001 × 643 × 1211

0.0

0.25

u/u∞
0.5

0.75

(a)

0.00252.5

1.0(a) 2.5y/δ00.0015
1.5

Cf

1.0

(b) 2.5

−5

0

5

10

0.00050.5

1.0-0.0005
−0.5

y/δ0

0.00

−5

0

5

10

(b)

2.5
2.0

1.0

-5
−5

00

55

10
10

-10
−10

-5
−5

00

55

10
10

(x − xsh) /δ0

2.6
2.6

2.2
2.2

−5

0

(x − xsh) /δ0

5

Figure 2. Mean streamwise velocity distributions.
Quasi-adiabatic; (b), heated; (c), cooled.

10

pw /p∞

0.0
−10

-10
−10

2.4
2.4

y/δ0

(c)

0.001
1.0

2.0

0.0
−10

−3

0.002
2.0

2.0

0.0
−10

( × 10 )

(a),

2.02
1.8
1.8
1.6
1.6
1.4
1.4

2

1.2
1.2

1.01

robustly with the compact scheme (Kawai et al., 2010). Also,
the third-order TVD Runge-Kutta method is used to conduct
time integration explicitly (Gottlieb & Shu, 1998).
The flow and grid parameters of the present cases are
summarized in Tab. 1. The Reynolds number and grid resolutions are extracted from x = 12δ0 upstream of shock impingement location xsh . The interaction length Lint is obtained
as the length between shock impingement location xsh and theorigin of the reflected shock. Wall heating or cooling decided
by Tw /Tr affect the wall density and viscosity based on Eqs. (4)and (7), which changes friction Reynolds number Reτ significantly between the cases, compared to Reynolds number ofboundary-layer momentum thickness Reθ . In the streamwise
and spanwise direction, the uniform grids are employed. In
the wall-normal direction, the grid spacing is stretched fromthe wall up to the outer boundaries, where the grid growth rate
is approximately 1.03 at the inner boundary layers. The spatial
resolutions of the computational grids are comparable to thoseused in the prior work of DNS (Bernardini et al., 2016).

(x − xsh) /δ0

Figure 3. Streamwise distributions of mean (a) skin frictionand (b) wall pressure. Black, quasi-adiabatic; red, heated;
blue, cooled.

and cooled cases, respectively. The results in Fig. 2 indicatethat wall cooling is to the benefit of preventing flow separations, and the differences in the behaviors of shock-induced
separated flows appear in the streamwise distributions of wallquantities in Fig. 3. In Fig. 3 (a), the skin friction decreases byflow separations, and the distributions show lower and higher
values in the heated and cooled cases, respectively. Also, twopeaks are observed for each wall temperature case; the peaks
become sharply as wall temperature decreases, and the downstream peaks are located at (x − xsh )/δ0 ≈ −2 regardless of
the wall temperatures. The prior study reported that the existences and sharpness of the upstream peak are dependent on
the Reynolds number (Morgan et al., 2013), therefore, the profiles in Fig. 3 (a) are considered to involve the effects of the
Reynolds number differences shown in Tab. 1. In Fig. 3 (b), thestreamwise gradients of pressure jump at the starting point ofboundary layer separations get smaller and larger in the heatedand cooled case, respectively. The pressure jump correspondswith a reflected shock that generates low-frequency wall pressure fluctuations, as discussed later in Fig. 7.

RESULTS
Mean flow fields affected by wall temperature
Figure 2 shows the mean velocity distributions of thepresent cases. The streamwise location is represented as thenon-dimensional distance from shock impingement location
xsh . Shock-induced separations are observed in all the cases;
the separated regions grow in the heated case and shrink in thecooled case, compared with the adiabatic case. Also, the separation points move upstream and downstream in the heated3

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

10

−4

PSD (p′w)
−3

10

10

−2

0.016
1.6

−1

10

2
p′w p′w /p∞

St = fδ0 /u∞

0.012
1.2

100

10−1

St = fδ0 /u∞

−0.5

0

0.5

1.0

0.00

-0.5
−0.5

00

(x − xsh) /Lint

0.5
0.5

1
1.0

Figure 5. Distributions of wall pressure fluctuations divided
into the components in Eq. (10). Solid, LHS; dashed, 1st termof RHS; dotted, 3rd term of RHS. Black, quasi-adiabatic; red,heated; blue, cooled.

10−2

St = fδ0 /u∞

-1
−1.0

6

10−1

100

0.006
0.6

0.002
0.2

−1.0

100

101

0.01
1.0
0.008
0.8

0.004
0.4

10−2

(c)

−2

0.014
1.4

(a) 101(b) 101( × 10 )

−1.0

−0.5

0

0.5

1.0

and the root-mean-square can be obtained as,

10−1p′ p′ = p′lf p′lf + 2p′lf p′hf + p′hf p′hf .

(10)

−2

10

−1.0

−0.5

0

(x − xsh) /Lint

0.5

1.0

Note that the cross-term of low- and high-frequency quanti5
ties are small enoughto be ignored. Figure 5 shows streamwise distributions of wall pressure fluctuations corresponding
to the terms in Eq. (10). The solid lines consisting of all
frequency components (LHS in Eq. (10)) show the peak at(x − xsh ) /Lint ≈ −0.8 that gets larger as wall temperature decreases. At the same streamwise location, the low-frequency
components (1st term of RHS in Eq. (10), dashed lines) also
show the peaks, although the high-frequency components (3rdterm of RHS in Eq. (10), dotted lines) increase smoothly without the remarkable peak. The results indicate that the maximum peak of wall pressure fluctuations consists of the lowfrequency components. Also, comparing the wall temperaturecases, a higher peak value of the low-frequency components
is observed in the cooled case than in the quasi-adiabatic andheated cases. The results suggest that wall cooling has the
disadvantage of providing strong low-frequency wall pressure
fluctuations, although the flow separations are prevented compared with quasi-adiabatic and heated wall cases.
Finally, flow phenomena corresponding to the low- andhigh-frequency wall pressure fluctuations are detected. Instantaneous fields of streamwise velocity and pressure in thequasi-adiabatic case, consisting of the low- or high-frequency
fluctuations as Eq. (9), are shown in Fig. 6. Regarding thestreamwise velocity distributions (left columns), flow fields,
including low-frequency fluctuations, in Fig. 6 (c) show overall structures of boundary layer separations. On the other hand,fine turbulent structures can be seen in the flow fields, including high-frequency fluctuations, in Fig. 6 (e). In the pressure
fields (right columns), the fluctuations of separated shear layers are obtained as the high-frequency components in Fig. 6(f), compared to the low-frequency components in Fig. 6 (d).
The results suggest that high-frequency fluctuations of the flow
and pressure fields are associated with turbulent structures ofthe separated shear layers. Also, low-frequency fluctuations
seem to contain large-scale motion of overall boundary layer
separations.
To discuss the flow physics associated with low- and highfrequency fluctuations more statistically, the frequency components of streamwise velocity and pressure fluctuations are

Figure 4. Power spectral density distributions of wall pressure. (a), Quasi-adiabatic; (b), heated; (c), cooled.

Wall temperature effects on low frequency fluctuations
Figure 4 shows premultiplied power spectral density distributions of the wall pressure as a function of streamwise location and Strouhal number St = f δ0 /u∞ . The streamwise
distances from shock impingement location xsh are scaled byinteraction length Lint of each wall temperature case to line
up the reflected shock locations. The spectral components are
non-dimensionalized by the integral over the entire frequency
ranges for each streamwise location. In Fig. 4, the spectral distributions roughly can be divided into two components: lowand high-frequency components. Regarding() low-frequency
components, spectral density of St ∼ O 10−2 is observed at(x − xsh ) /Lint ≈ −1.0 in all the cases. In high-frequencyre( )gions, dominant spectral distributions at St ∼ O 100 are observed throughout the streamwise locations. Also, the overall
spectral distributions shift to lower and higher frequencies in
the heated and cooled case, respectively.
Next, contributions of the low- and high-frequency components to mean wall pressure fluctuations are evaluated.
By performing the inverse Fourier transformation, instantaneous pressure fluctuations are divided into the low- and highfrequency components as,p′ = p′lf + p′hf ,(9)where the subscripts lf and hf denote quantities consistingof low- and high-frequency components, respectively. The
boundary of low- and high-frequency is decided as St =f δ0 /u∞ = 0.05, where f is the frequency. The pressure fluctuations in Eq. (9) are averaged in time and spanwise direction,4

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

0.0

0.25

0.5

0.75

p/p∞, plf /p∞, phf /p∞

1.0

1.0

2.0

y/δ0

−1.0

−0.5

0

0.5

1.0

−1.0

−0.5

0

0.5

1.0

−1.0

−0.5

0

0.5

1.0

−1.0

−0.5

0

0.5

1.0

y/δ0

2.0

1.0
−1.0

−0.5

0

0.5

1.0

(f )

y/δ0

2.5
2.0

1.0
0.0

0.0

(d) 2.5

1.0

(e)

3.0

1.0

2.5
2.0

0.0

2.5

2.0

1.0

(c)

2.0

(b)2.5(a) 2.5

0.0

1.5

0.0

2.5
2.0

1.0
−1.0

−0.5

0

(x − xsh) /Lint

0.5

1.0

0.0

(x − xsh) /Lint

Figure 6. Instantaneous flow fields of streamwise velocity (left columns) and pressure (right columns) in the quasi-adiabatic case,consisting of low- and high-frequency fluctuations as Eq. (9). (a,b), original instantaneous flow fields; (c,d), flow fields consisting of
5
low-frequency fluctuations; (e,f), flow fields consisting of high-frequency fluctuations.

extracted as,∫ fcri)(q′ q′ lf =P( f )d f ,

(11)

(
)
q′ q′ hf =(12)turbulent structures in separated shear layers also can be affected by the semi-local Reynolds number because separated
shear layers originate from the separation of incoming turbulent boundary layers. Thus, the changes of friction Reynolds
number by wall heating and cooling, shown in Tab. 1, can beone of the causes of the high-frequency spectral distribution
shifts observed in Fig. 4.

fmin∫ fmaxP( f )d f ,fcriwhere q is the flow quantities of streamwise velocity or pressure and P is the power spectral density of corresponded flow
quantities. The subscripts min and max denote the minimum
and maximum frequency of used time-stepping data, respectively. Criterion frequency fcri is set to St = fcri δ0 /u∞ = 0.05
that is the same boundary used in Eq. (9). Figure 7 shows
distributions of streamwise velocity (left columns) and pressure (right columns) fluctuations, extracted by Eqs. (11) and(12), in the quasi-adiabatic case. In Fig. 7 (a) of low-frequency
streamwise velocity fluctuations, overall flow separations arehighlighted. Also, in Fig. 7 (b) of low-frequency pressure
fluctuations, impinging and reflected shock are observed. Especially in the reflected shock, the distributions extend to thelower wall at streamwise location (x − xsh ) /Lint ≈ −1.0, suggesting that low-frequency components of wall pressure fluctuations shown in Figs. 4 and 5 are associated with the reflected shock motion. As shown in Fig. 3 (b), the pressure
gradients of reflected shock change between the wall temperature cases, which can be considered to affect the differencesin low-frequency wall pressure fluctuations in Fig. 5. On theother hand, the high-frequency fluctuations in Figs. 7 (c) and(d) distribute at separated shear layers, which are consistent
with the observation in Fig. 6. Also, near-wall turbulence at attached boundary layer regions is highlighted in Figs. 7 (c). The
results suggest that turbulent structures at attached boundary
layers and separated( ) shear layers provide the spectral components at St ∼ O 100 in Fig. 4. Regarding near-wall turbulence
at attached turbulent boundary layers, a previous study showsthat the length scales of turbulent structures in thermal turbulent boundary layers depend on the differences in the semilocal Reynolds number, regardless of the temperature, density,and viscosity themselves (Hirai et al., 2021). Furthermore,

CONCLUSIONSIn this study, wall heating and cooling effects on lowfrequency dynamics in the shock-wave and turbulent boundary
layer interactions were investigated by direct numerical simulations. Power spectral density of the wall pressure showedthat the low-frequency fluctuations happen locally at the starting point of boundary layer separations, although the highfrequency components distribute throughout the streamwise
locations. By dividing the root-mean-square of wall pressure
into the low- and high-frequency components, large contributions of the low-frequency components to the maximum peakof wall pressure fluctuations were elucidated. The peak of
the low-frequency wall pressure fluctuations gets larger in thecooled wall cases, suggesting that wall cooling has some undesirable contributions to decreasing the low-frequency wallpressure fluctuations that can lead to structural fatigue. Furthermore, to detect flow physics associated with the lowand high-frequency wall pressure fluctuations, the corresponding frequency fluctuations of streamwise velocity and pressure fields were observed. Distributions of the low-frequency
streamwise velocity and pressure fluctuations indicate that
overall boundary layer separations and reflected shock are associated with the low-frequency wall pressure fluctuations.
Regarding the reflected shock, differences in pressure gradients are observed between the wall temperature cases, which
can change low-frequency wall pressure fluctuations. Also,
high-frequency fluctuations of streamwise velocity and pressure distribute at separated shear layers and near-wall regionsof attached boundary layers. The semi-local Reynolds number effects on near-wall turbulence, reported in a previous
study, are considered to affect the frequency shifts of wall5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(u′u′)lf /u∞, (u′u′)hf /u∞
2

0.0

0.5

(p′ p′)lf /p∞, (p′ p′)hf /p∞

2

1.0

1.5

(a) 2.5

2.0

2( × 10 )
−2

2.0

0.0

−2

y/δ0

1.0
−1.0

−0.5

0

0.5

2.5
2.0

1.0

0.0

(d) 2.5

−1.0

−0.5

0

0.5

1.0

−1.0

−0.5

0

0.5

1.0

2.0

1.0
0.0

( × 10 )

2.0

y/δ0

(c)

2

5.25 7.0

3.5(b)2.5

1.0
0.0

1.75

1.0
−1.0

−0.5

0

(x − xsh) /Lint

0.5

1.0

0.0

(x − xsh) /Lint

Figure 7. Distributions of streamwise velocity (left columns) and pressure (right columns) fluctuations in the quasi-adiabatic case,obtained by Eqs. (11) and (12). (a,b), low-frequency components of Eq. (11); (c,d), high-frequency components of Eq. (12).

pressure spectral distributions at high-frequency regions. In
future work, we will investigate clear contributions of pressure
gradients of reflected shock and semi-local Reynolds number,which are changed by wall heating and cooling, to low- andhigh-frequency wall pressure fluctuations, respectively.

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ACKNOWLEDGEMENTS
This work was supported in part by Japan Society for
the Promotion of Science Grant-in-Aid for Scientific Research(B) KAKENHI 21H01523. A part of this research used
computational resources of the Fugaku computer provided by
the RIKEN Advanced Institute for Computational Science.
(Project ID: hp210099 and hp220034).

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