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

DIRECTED PERCOLATION PHENOMENA IN SUBCRITICAL TRANSITION
OF HIGH-ASPECT-RATIO DUCT FLOW

Kazuki Kohyama
Department of Mechanical Engineering
Tokyo University of Science
2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan7521701@ed.tus.ac.jp

Masaki Sano
Institute of Natural Sciences, School of Physics and Astronomy
Shanghai Jiao Tong University
800 Dongchuan RD. Minhang District, Shanghai, 200240, China
sano.masaki@sjtu.edu.cn

Keiichi Tamai
Takahiro Tsukahara
Research Center for Computational Design
Department of Mechanical Engineeringof Advanced Functional Materials
Tokyo University of Science
National Institute of Advanced Industrial Science and Technology 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan
1-1-1 Umezono, Tsukuba, Ibaraki, 305-8568, Japantsuka@rs.tus.ac.jp
k-tamai@aist.go.jp

ABSTRACT

laminar and turbulent regions are arranged alternately and intermittently with randomness only in the streamwise direction.
Each turbulent puff may split into two turbulent puffs or decay and, as the Reynolds number goes down to the globalstability criticality, those two events are stochastically comparable (Avila et al., 2011; Mukund & Hof, 2018).
Recently, these turbulent branching and some statistics of
STI were shown to have directed percolation (DP) universality,
as conjectured by Pomeau (1986). A narrow Taylor-Couette
flow (TCf) and a wide-gap annular flow exhibit similar (1+1)dimensional STI of pipe flow and follow (1 + 1)-D DP universality class, where the first number is the spatial dimension andthe latter indicates time (Lemoult et al., 2016; Takeda et al.,2020). In other canonical flows in wide channels, the STI is
more complicated because a spatially two-dimensional pattern
is formed with turbulent stripe and oblique band (Prigent et al.,
2002; Barkley & Tuckerman, 2005; Tsukahara et al., 2005;
Duguet et al., 2010; Tuckerman & Barkley, 2011; Tuckerman
et al., 2020). Some subcritical transitions in the wide channels
were found to follow (2 + 1)-D DP universality class; the Waleffe flow by Chantry et al. (2016), the wide TCf by Klotz et al.
(2022), and the channel flow (also known as plane Poiseuilleflow, PPF) by Sano & Tamai (2016); Takeda & Tsukahara(2019); Shimizu & Manneville (2019).
For the PPF between two parallel infinite plates, the localized turbulence forms the stripe pattern consisting of oblique
bands for Re < 1500 (Tsukahara et al., 2010), despite the linear
stability analysis by Orszag (1971) saying that ReL = 5772.2.
The experimental nominal-channel flow and the numerical
PPFs with large-scale domains show good agreements with
(2 + 1)-D DP universality class, but the transition profiles below the criticality (Re < 1000) deviates from that, where theflow can maintain turbulence for Re ≈ 800 with isolated band
which is (1+1)-D STI and straight ahead at an angle of approximately 45◦ (Tao et al., 2018; Xiao et al., 2021). The presenceof spanwise sidewalls, i.e., a duct as experimental channel, affects the intermittency formation. This sidewall and finite-size
effects depend on the duct geometry such as the aspect ratio
A defined in the flow-cross-section as a ratio of the long sideagainst the short one. In a square duct of A = 1, (1 + 1)-D STI

Directed percolation (DP) and subcritical turbulent transitions in channel flow, which have been a debatable issue in recent years, are investigated by direct numerical simulations forhigh-aspect-ratio channel flows with spanwise sidewalls, i.e.,very thin, wide duct flows. The presence of sidewalls in theduct flow prevented sustained stable localized turbulence bandat low Reynolds numbers (700 ≲ Re ≲ 1000), which allowed
global turbulence maintenance in the infinitely wide channel
flow, and the critical values were approximately Rec = 1000.
Localized turbulence was spatio-temporally intermittency, andits split and decay were generally similar to those in the channel flow, although there were some behaviors specific to theduct flow. Long-term Lagrangian observation of the split anddecay of localized turbulence near the criticality showed thebranching of turbulence in the space-time-diagrams. The spatial dimension of the branching depended on the duct geometry, with a spatial one-dimensional spread when the distance
between sidewalls was not wide enough (like in a square duct),
but two-dimensional when it was wide, qualitatively confirming (1 + 1)-dimensional and (2 + 1)-D DP characteristics, respectively. The power-law scaling exponents for the Reynolds
number dependence of the transition profiles were found to beconsistent with the specific critical exponents of the two typesof DPs, respectively.

INTRODUCTION
Wall-bounded shear flow in subcritical regime forms alarge-scale intermittency of coexisting laminar and turbulent
flow, even when the Reynolds number Re is much lower thanthe upper critical value ReL determined by the linear stability.
Also, once the flow becomes completely laminar, turbulence
does not spontaneously nucleate, which is know as an absorbing phase transition. That is turbulence can be sustained globally by maintaining equilibrium localized structures. The localized turbulence generally has a large-scale spatio-temporalintermittency (STI). This phenomenon has been found to beuniversal at the onset of the subcritical turbulent transition. A
well-known example is the turbulent puff in pipe flow, where112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. Computational conditions of medium- to high-aspect-ratio duct and plane Poiseuille flows (PPF). Case shows an aspect ratioof duct channels A (≡ Lz /Ly ), but A∞ indicates the infinite channel (PPF) with periodic boundary condition in spanwise direction. Li∗
is the domain size non-dimensionalized by the channel half width δ , and Ni is the number of grids. The wall-unit resolutions ∆+ arevalues at Reτ = 64, but at Reτ = 48 for A∞ with largest case.

Case

Lx∗ × Ly∗ × Lz∗

Nx × Ny × Nz

A24

512 × 2 × 48

4096 × 64 × 1016

A96

512 × 2 × 192

4096 × 64 × 4088

512 × 2 × 409.6
819.2 × 2 × 1024

A∞

Reτ

∆x+

∆y+

∆z+

46–1108.0

0.2–4.6

3.04096 × 64 × 4096

49–1108.0

0.2–4.6

6.44096 × 64 × 8192

37–49

9.6

0.2–3.5

6.0

shows the same phenomena as a turbulent puff in a pipe flow
(Khan et al., 2021). At A = 9, Takeishi et al. (2015) identified
two-dimensional zigzag pattern like a stripe structure. The infinite/finite size effect might be a main cause of the modulationin DP features and of the critical values discrepancy.
In the present study, we performed direct numerical simulations (DNSs) of medium-to-high-aspect-ratio duct flows, up
to A = 96, and the PPF in subcritical regime at low Reynolds
numbers, to discuss the sidewall effect on DP features in turbulent branching and the transition profiles.

h = 2δ
y, v

Lz
z, w
Figure 1. Schematic of rectangular duct flow. Boundary geometry is defined by the aspect ratio A ≡ Lz /Ly (= Lz /2δ ) > 1.

Computations
The non-dimensional governing equations for the velocity
ui = (u, v, w) and the pressure p are the equation of continuity
and the Navier–Stokes equation for incompressible Newtonian
fluid in the Cartesian coordinate system xi = (x, y, z):

NUMERICAL SET-UP AND METHODOLOGY
Geometry of Duct Flow
The duct flow is considered as the flow in the channel
between two parallel planes with two parallel sidewalls in the
spanwise direction to be driven by a constant pressure gradient(dP/dx) in the streamwise (x) direction. The typical configuration is depicted in Fig. 1 with the three-dimensional Cartesian
coordinates system. The geometry of the rectangular duct isdetermined by the distance between the spanwise sidewalls,
that is, we used the width-to-height aspect ratio defined as
A≡

Lz
Lz
=
Ly
2δ

Lx

x, u

∂ u+
i
=0
∂ xi∗
∂ u+
∂ u+
∂ p+
1 ∂ 2 u+
i
i
i
+ fi
+ u+j
=− ∗ +
∗
∗
∂t
∂xj
∂ xi
Reτ ∂ x∗2
j

(2)
(3)

where the superscripts of ∗ and + indicate quantities nondimensionalized by δ and the friction velocity uτ , respectively. fi in the last term of Equation (3) is a body force tomimic the sidewall solid phase in the channel flow by meansof the immersed boundary method (IBM). The control parameter of the present DNS is the friction Reynolds number Reτ
normalized by uτ converted
p from the constant pressure gradient: Reτ ≡ δ uτ /ν = δ −dP/dx∗ /ν. Here, the pressure
P is already divided by the fluid density ρ, and ν represents
the kinematic viscosity. Given a sufficiently large
p A, the friction velocity asymptotically approaches uτ = τw,y /ρ withthe wall shear stress τw,y on the main wall surface at y = 0 and
2δ . In the following discussion, we use the Reynolds number
Re defined with the streamwise velocity at the center line inlaminar flow uc lam. = uτ Reτ /2 as(1)

where δ is the channel half width in the wall-normal (y) direction. A typical example is the square duct flow with A = 1. A
rectangular duct with A > 1 has a wider distance between the
spanwise sidewalls, as shown in Fig. 1. In all cases, the streamwise velocity u(y, z) is not uniform in the spanwise direction
because of sidewalls and corners, unlike PPF. However, theseeffects attenuate for high-aspect-ratio. The detailed solution isin Wu & Song (2022); Kohyama et al. (under review).
In the present DNS study, we chose two different valuesof A = 24 and 96 to cover medium- and high-aspect-ratios.
This range allows us to exhaustively investigate the sidewall
effects on the turbulent flow and spatio-temporal intermittent structures. For reference, previous experimental nominalchannel (duct) flows in the subcritical region have been performed comprehensively with a wide range of aspect ratios
A = 7.5 by Lemoult et al. (2012), 52 by Seki & Matsubara(2012), 83 by Yimprasert et al. (2021), 180 by Sano & Tamai(2016), and 245 by Paranjape (2019). However, since their
the streamwise domain is finite, it has been difficult to discuss statistical steady states about their transition profiles andturbulent branching near the criticality. Except for the reportof Reg = 1500 for a narrow duct with A = 7.5 (Lemoult et al.,2012), many experiments reported Reg ≈ 1000 or less for large
values of A. Numerical studies have already confirmed a similar trend only for duct flows with low- to medium-aspectratios (A = 1–9, 50), but not for higher one (Khan et al., 2021;
Takeishi et al., 2015; Wu & Song, 2022).

Re ≡δ uc lam.
= Re2τ /2
ν(4)

We adopted two types of numerical boundary conditions:the non-slip and the periodic boundary conditions. The nonslip condition was adopted on the main walls normal to the y
direction, while the periodic boundary conditions were in the x
direction. The non-slip boundary on the spanwise sidewall was
configured by the direct-forcing IBM around the spanwise end
faces of the computational domain (Fadlun et al., 2000). We
applied the following IBM rules to the staggered-grid velocity
components at the sidewall boundary: no inflow or outflow at
the sidewall surfaces, and u = 0 inside the sidewalls.
Equations (2) and (3) were spatially discretized using finite differences and with fine enough grid resolutions ∆xi+ , aslisted in Table 1, according to the standard criteria of DNSs
(Tsukahara et al., 2005). We adopted the uniform grid in the x2

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

Figure 2. Typical instantaneous flow field and space-time-diagrams (STDs) in duct flow with A24 for Re = 1128. (a) Snapshot of
instantaneous intermittency with turbulent spots and oblique bands visualized by the wall-normal velocity v+ in the channel central
x–z plane at y = δ . The sidewalls are at z/δ = 0 and 48. (b–f) STDs of |v+ (x̂ or z, t)| as the spatio-temporal intermittencies whichindicate localized turbulence blanching. Localized turbulences are shown in black and laminar flow in white. In (b,c,d), spaces are thestreamwise direction at near the sidewall to the center between the sidewalls as on the dashed lines z/δ = 6, 12 and 24 in (a). Here, thespace x̂ (= x −U f · t) is in Lagrangian coordinate corrected with the frame speed U f = 15.5uτ . In (e,f), space is taken in the spanwise
direction at x̂/δ = 128 and 384. (g) typical STD of (1 + 1)-dimensional directed site percolation (DP).

Spatio-Temporal Intermittency

and z directions and the non-uniform in y, where the grid wasfiner near the main wall, cited from Moin & Kim (1982). For
the present DNS of duct flows, the spanwise grid sizes were
finer than those for PPF, especially because of the boundary
layer on the spanwise sidewalls. Table 1 lists the parameters
for the rectangular duct flows and the PPF. The time advancement was carried out using the second-order Crank–Nicolson
and Adams–Bashforth schemes for the viscous term in the y direction and the other terms, respectively. Further details about
the numerical methods used can be found in Abe et al. (2001).

Figure 2(a) and 3(a) show typical instantaneous flow
fields in both ducts in the x–z plane at the mid-channel of y = δnear the criticality for different aspect ratios. High fluctuations of the wall-normal velocity v+ in red and blue indicate
longitudinal streaks and fine vortices as turbulence. To summarize briefly, the intermittencies depend on the spanwise domain size, as shown below. The duct flow with A24 does notform a perfect one-dimensional intermittency like a puff, but
forms an quasi-1D structure in the streamwise direction consisting of turbulent spots and short oblique bands. For A96, thespot is not stationary, but some growing bands are dominant,and leads to a 2D structure similar in PPF. For the details of
their stochastic behaviors, see Kohyama et al. (under review).
Localized turbulence is usually advected in the streamwise direction with bulk velocity um , since there is positive
mean flow in the PPF and duct flow. STDs represent thetime evolution of turbulent and laminar flows at the spacemarked by the dashed lines in Figs. 2(a) and 3(a) from a Lagrangian view. For example, the horizontal axes of STDs in
Figs. 2(b-d) and 3(b-e) are the streamwise space, but observed
from a Lagrangian coordinate (i.e., x 7→ x̂ ≡ x − U f · t). Here,the frame velocity U f is slightly larger than the bulk velocity um in the critical neighborhood focused on in this study(U f > um ≈ 14.5uτ ). Because a turbulent band’s head grows

RESULTS AND DISCUSSIONSWe discuss the spatial spread of the spliting and decaying
behavior of localized turbulence in duct flows with the A = 24
and 96 from the space-time-diagrams (STDs). Its dependence
on the distances from the sidewall surface and the aspect ratio are also discussed. Subsequently, based on the DP scaling
theory, the critical exponents of the transition profiles such as
the Reynolds number dependence of the turbulent fraction that
follow a power law near the criticality are assessed in reference to the unique exponent values of DP universality classes.
For reference, DP scaling was also performed for that of the
PPF. The differences from the duct flow in terms of the criticalvalues, the exponents and its fitting range were investigated.

3

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

Figure 3. Typical instantaneous flow field and STDs in duct flow with A96 for Re = 1105. (b-g) STDs presented in the same manneras in Fig. 2, but the frame speed is U f = 15.25uτ . (h) typical STD of (2 + 1)-D site DP for one spatial direction in the 2D space.

obliquely and extends to the downstream faster than um . Note,in the beginning of the intermittency away from the criticality,e.g., when a robust stripe pattern is formed, U f and um are approximately the same. For details of STD in a frame moving
Lagrangian view, see Kashyap et al. (2020), but in the PPF.
Subsequently, the branching structure of the localized turbulence is compared with the STDs of (1 + 1)-D and (2 + 1)-Dsite DP shown in Figs. 2(g) and 3(h), respectively, which isone of the models of directed percolation. The details about
the DP model can be found in Henkel et al. (2008). Then,we will qualitatively estimate the dependence on the distancefrom the sidewalls with respect to the presence or absence ofDP-like features in the spatial spread of the branching.
For the STI of |v+ (x̂,t)| in the narrow duct flow with A24,
the STD at the center of the sidewalls (z = 24δ ) shown in
Fig. 2(b) is in a good qualitative agreement with the STI of(1 + 1)-D DP shown as an example in Fig. 2(g). However, theSTIs at z = 6δ , 12δ near the sidewall shown in Fig. 2(c,d)
do not have a single series of branches like the feature of(1+1)-D DP. The STDs of |v+ (z,t)| in the span space shown in
Fig. 2(c,d) are similarly strongly influenced by the sidewalls,
each having only independent events: band and spot attachment, decay, and reflection at the sidewalls. It indicates thatthe DP phenomenon is not effective in the spanwise direction

in cases where the domain is so narrow that turbulence does
not split in the spanwise direction. As shown in Fig. 3(b,c),the both STDs at z = 48δ and 96δ as far away from the sidewall are qualitatively almost the same for the wide duct flow ofA96, indicating that the sidewall effect is not significant. Thisbranching is also two-dimensional based on the STDs in the
spanwise direction at x̂ = 128δ and 384δ shown in Fig. 3(f,g).
These branches are two-dimensional, which are different from
Fig. 2(b), and qualitatively similar to the STD of (2 + 1)-D DP
shown in Fig. 3(h), although slightly different. Closer to thesidewall (z = 6δ and 24δ ), as shown in Fig. 3(d,e), the splitting behavior of the developed localized turbulence becomes
weaker. Thus, STI is affected by the sidewalls and STD depends on its location, i.e., it is actually not characterized bybranching because of turbulence trapping on the sidewalls.
In summary, the stochastic behaviors of bands, including
the splitting, colliding, and decaying, appear as a branchingprocess, as illustrated in the STDs of localized turbulence. In
Fig. 2(b), the duct flow of A24, which is strongly affected by
the sidewall and the finite size effect, forms a spatially onedimensional branching process similar to (1 + 1)-D DP STI,in spite of asymmetry in the streamwise direction. This is nottrue for A96 (Fig. 3(b,c)), where we should consider a spatially
two-dimensional intermittency like in the infinite channel flow

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 2. Critical Reynolds number Rec and the critical exponent βfit expected by Equation (5) for each fitting range. Also
shown is the global critical Reynolds number Reg , which is
the lower limit of the globally sustained turbulence, cited from
Kohyama et al. (under review). For reference, the theoreticalcritical exponents β of the two types of DPs are listed below.

(PPF). We may conjecture that DP universality changed from
(1 + 1)-D to (2 + 1)-D between A24 and A96. On anothernote, to reproduce more exactly the STD such as asymmetryor fractal-like branching, it will be necessary to adopt the DPbased model proposed by Manneville & Shimizu (2020) or the
Domany-Kinzel (DK) model, which generalized DP model.

Case

Reg

Rec

β

Fitting range

Transition Profiles and Critical Exponents

A24

1105

1119

0.28 ± 0.02

1100–1600

The transition profile can be described as the Reynolds
number dependency of the order parameter from a developed
turbulence to fully-laminar state. For the order parameter
which indicates the macroscopic order in the overall system,
the turbulent fraction was adopted, which is the degree of spatial intermittency, i.e., the fraction of localized turbulence inthe total domain. The temporal turbulent fraction Ft was calculated using the wall-normal velocity in the x–z plane at thecenter of the main channel (y = δ ) as shown in Figs. 2(a) and
3(a), using the binarization of turbulent or laminar state withthe threshold as |v+ (x, δ , z)|θ = 0.1. Note that Ft = 0 corresponds to the fully-laminar, and Ft ≈ 1 the fully-developed
turbulent state. In the profiles, the turbulent fraction ⟨Ft ⟩t islong-term-averaged over Ft at the statistical steady state (atleast 500δ /uτ , i.e., about 15 wash-out times in the streamwise
domain), which is much longer than the time scale of the bandsplitting frequency, even near the criticality. In particular, thetime scale of the turbulence branching phenomenon becomessignificantly longer as the critical value is approached, so thetime-averaging is longer than ∆t ∗ = 1000 in the vicinity of thecritical value. Based on the transition profiles (⟨Ft ⟩t vs. Re)of two duct flows and the PPF, we investigated the presenceor absence of DP universality class’ scaling laws in power-like
transition, their critical values and exponents, and the global
critical Reynolds number Reg , which is the lower bound forsustained turbulence. (Figure for profiles is not shown in thispaper, but is available in Kohyama et al., under review.)
Profiles of phase transitions are often followed by a powerlaw of the control parameter (in this study, the Reynolds number). The present profiles in the near criticality similarly agreewith the following power-law scaling as

A96

1058

999

0.57 ± 0.03

1100–1405

A∞

722

994

0.58 ± 0.19

1155–1405(1 + 1)-D DP——∼ 0.276—(2 + 1)-D DP——∼ 0.583—⟨Ft ⟩t ∼Re − RecRec(1982) that three-dimensional disturbances grow to turbulence
at Re > 1000. While, for the duct flow with A24, the criticalvalue is slightly higher (Rec = 1119), and the critical exponent
is lower (β = 0.28 ± 0.02) and in agreement with β(1+1)-D ,indicating a drastic transition to fully-laminar flow and sensitivity to changes in the Reynolds number around the criticality.
Note, the large error range of the critical exponent in the PPF isdue as the plots when the order parameter is close to ⟨Ft ⟩t = 0
are not near the critical value Rec and not included in the fitting, which is due to the peculiar PPF profile. Due to the characteristic turbulence-maintaining structure at 700 ≲ Re ≲ 1000
in PPF, the profile at Re < 1155 deviates from the second-orderphase transition. Where, the fitting ranges are different, asshown in Table 2, but generally 1100 ≲ Re ≲ 1500 as near
the critical values. However, above this Reynolds number region, all profiles are the same and the curve becomes gradualand deviates from the critical exponent of DP. Because around
Re = 2000, the intermittent structure is a pattern-structured turbulent stripe, which has robustness and no DP-like active splitting. It is suggested that all duct transitions with A ≳ 96 follow(2 + 1)-D DP, whereas (1 + 1)-D DP for not so high-aspectratio due to the formation of quasi-one-dimensional turbulent
spots like the turbulent puff in the transitional pipe flow.

CONCLUSIONβ

In the present study, directed percolation (DP) phenomena in the STIs of localized turbulence and there transition
profiles on subcritical regime have been investigated in two
high-aspect-ratios (A = 24 and 96) duct flows and the infinitely wide channel flow (PPF) by DNS to characterize its
phenomenon. Since the structure and behavior of intermittency become larger near the criticality, the previous experimental and numerical studies have not yet been able to capture
the essence of the critical phenomena due to insufficient domain and statistics. Long-term Lagrangian observations of theinteraction of multiple localized turbulences were performedin the present DNSs for the largest-scale domain.
For the effects of the presence or absence of sidewalls andthe distance between sidewalls (i.e., duct flow or PPF, and aspect ratio) on the spatio-temporal intermittencies (STIs) andthus the transition profiles, the following points were found.

(5)

where Rec is the critical Reynolds number of profile following the power law and β is the critical exponent with positive value. The critical Reynolds numbers Rec and the criticalexponents β of the duct flows and the PPF transition profiles
are summarized in Table 2, calculated by scaling according
to Equation (5). The global critical values Reg as cited from
Kohyama et al. (under review) are also noted. In general, Recdiffers from Reg due to the domain size effects: e.g., Rec = 994
and Reg = 722 in the PPF of A∞. However, in the present ductflows, the difference between the two critical values is not sosignificant, indicating a “pure” second-order phase transition.
In terms of the DP universality and the scaling theory,
β in Equation (5) has a specific value depending on the spatial dimension of the intermittency. For example, if turbulent
spots spread one-dimensionally, we expect β(1+1)-D ∼ 0.276,
whereas β(2+1)-D ∼ 0.583 if two-dimensionally. The critical values for both the duct flow with A96 and the PPF corresponding to A∞ were about Rec = 1000, and both critical exponents agreed with β(2+1)-D within the error range.
Also, the critical value for wide duct flow asymptotically approaches Rec ≈ 1000, which is consistent with the expectation in the nominal-channel flow experiment in Carlson et al.

— The global turbulence maintenance limitation is approximately asymptotic to Reg = 1000 even for high-aspectratio duct flows with spatial finiteness, as predicted on theexperimental channel flow in Carlson et al. (1982). It does
not maintain turbulence down to Reg = 722 as in the PPF,
and for Re < 1000 all turbulent spots and bands decay
within a finite time, becoming fully-laminar flow.
— Local turbulence split and decay are significantly ob-

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
served in duct flow near the criticality (Re ≈ 1100–1200),
causing large-scale STI as a single series of branches.
For medium aspect ratio (A24), turbulent spots and bands
form quasi-one-dimensional intermittency only in thestreamwise direction at the center of the sidewalls, closely
similar to (1 + 1)-D DP branching. While, for high aspectratio (A96), the intermittency is two-dimensional in thestreamwise and spanwise directions, and the branching isclose to (2 + 1)-D DP. However, near the sidewalls, localized turbulent is not actively splitting, but is often decaying, etc., and its branching has no DP-like features.
— The Reynolds number dependency of the turbulent fraction (transition profile) follows the scaling of the powerlaw of DP around the critical value. The profiles of theduct flows with A24, and A96, and the PPF at Re ≈ 1100
were consistent with the critical exponents of (1 + 1)-Dand (2 + 1)-D DP universality classes, respectively.

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16.

For future work, a detailed discussion of DP universality
classes would require a quantitative investigation of the criticalexponents by power law scalings, not only for the transition
profile, but also for the laminar gap distributions of the STI.
Present numerical simulations were performed on the SXACE and SX-Aurora TSUBASA supercomputers with technical support from the Cyberscience Center at Tohoku University. This work was funded by Grant-in-Aid for the JSPS
(Japan Society for the Promotion of Science) Fellowships(grant number: JP19H02071 and JP22J14598). The first author (K.K.) expresses appreciation for the Fellowship Programfor Material Science Researcher at Tokyo University of Science, which is supported by MEXT (the Ministry of Education, Culture, Sports, Science and Technology).

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