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

INTER-SCALE TRANSFER OF PASSIVE SCALAR IN GRID TURBULENCE

Muyang Wang
Takuya Yurikusa
Department of Mechanical Systems Engineering Department of Mechanical Systems Engineering
Nagoya University
Nagoya University
Furocho, Chikusa-ku, Nagoya 464-8603, Japan
Furocho, Chikusa-ku, Nagoya 464-8603, Japanwang.muyang.k9@s.mail.nagoya-u.ac.jp
tyurikusa@i.softbank.jp

Koji Iwano
Yasuhiko Sakai
Department of Mechanical Systems Engineering Nagoya Industrial Science Research Institute
Nagoya University
Sakae, Naka-ku, Nagoya 464-0008, Japan
Furocho, Chikusa-ku, Nagoya 464-8603, Japanysakai@mech.nagoya-u.ac.jp
iwano@nagoya-u.jp

Yasumasa Ito
Department of Mechanical Systems Engineering
Nagoya University
Furocho, Chikusa-ku, Nagoya 464-8603, Japanyito@nagoya-u.jp

ABSTRACT
Direct numerical simulations were carried out to study the
turbulence generated by a regular square grid. Two-point correlation analysis using the scale-by-scale scalar (SBSS) equation and the third-order structure function was applied to reveal
the mechanism of inter-scale scalar transport. The results showthat the scalar displays a trend of inverse cascade in the vertical direction at the scale that is larger than the integral lengthscale.

U j +U j′!

∂ ⟨δ q2 ⟩ ∂ ⟨δ u j δ q2 ⟩ ∂ δU j ⟨δ q2 ⟩2
∂ Xj +
∂rj
∂rj
+
{z
}
|
{z
} | {z } |
4At
4Π
4ΠU
4A

∂ δUi
∂ δUi
2 ∂ ⟨δ u j δ p⟩−2⟨δ ui δ u j ⟩− ⟨ u j + u′j δ ui ⟩
−
∂
r
∂ Xj
ρ
∂
X
j
j
=|
{z
}
{z
}|
4P
4Tp

 


 2
′
2
∂
1 ∂2
∂ ⟨ uj +uj δq ⟩
2
−


2
∂ r2 2 ∂ X 2 
∂ Xj
2
+ν 
| {z j} + | {z j} ⟨δ q ⟩
|
{z
}
4Dν
4DX,ν
4Tu
"
!
!#
∂ u′i 2∂ ui 2
−2ν ⟨
⟩+⟨
⟩
∂xj
∂ x′j
|
{z
}
4ε
(1)
∂ ⟨δ q2 ⟩| ∂t{z } +INTRODUCTION
Since the first experiments of (Simmons & Salter, 1934),
the turbulent motion behind a grid has been investigated intensively in many experiments and numerical simulations. In
general, the turbulent flow behind the grid can be roughly classified into two regions: the developing region that wakes interact each other and are significantly affected by structuresof grids, and the developed region in the downstream areathat has quasi-homogeneous isotropic features. This difference
between upstream and downstream promotes the understanding of the development and evolution of turbulence. Therefore, many researchers have recently conducted a series ofstudies focusing on the extreme event, intermittency and selfpreservation on the upstream and downstream region.

Here, δ q2 = δ u2i , δUi = Ui − Ui′ and δ p = p − p′ . The angle bracket ⟨·⟩ means the ensemble average, Ui refers to the icomponent of mean velocity vector, and ui indicates the i component of the fluctuation of velocity vector, p is the pressure,respectively. The symbol δ represents a difference of the physical properties between the spatial location x = X + r/2 andx′ = X − r/2, where X is the centroid of two different points x
and x′ , r is the distance between the two points.
Similar to the energy transfer, the scale-by-scale scalartransport can be expressed as in Eq. 2 using the second-order
structure function (Hill, 2002):

On the other hand, the Karman-Howarth-Monin-Hill(KHMH) equation shown by Eq. 1 represents the inter-scaleenergy transfer between two spatial points x and x′ , and it has
been widely used to investigate the mechanism in various situations (Gomes-Fernandes et al., 2015; Portela et al., 2017;
Zhou et al., 2020).

1

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

∂ ⟨δ c2 ⟩ ∂ ⟨δ u j δ c2 ⟩ ∂ δU j ⟨δ c2 ⟩2∂ Xj +
∂rj
∂rj
+
{z
}
|
{z
} | {z } |
4At,c
4Πc
4ΠU,c
4Ac

∂ δC
∂ δC
−2⟨δ cδ u j ⟩− ⟨ u j + u′j δ c⟩
∂
r
∂ Xj
j
=|
{z
}
4Pc

 

 2

′
2
∂
1 ∂2
∂ ⟨ uj +uj δc ⟩
2
−


2
∂ r2 2 ∂ X 2 
∂ Xj
2
+ν 
| {z j} + | {z j} ⟨δ c ⟩
|
{z
}
4Dκ
4DX,κ
4Tc
"
!
!#
∂c 2
∂ c′ 2
−2κ ⟨
⟩+⟨
⟩
∂xj
∂ x′j
{z
}
|
4χ
(2)
∂ ⟨δ c2 ⟩| ∂t{z } +

U j +U j(b)

Figure 1. Schematic view of (a) the computational domainand (b) the turbulence-generating grid.

where C refers to the mean scalar , c indicates the fluctuationof scalar, and δC = C − C′ and δ c = c − c′ . This equation is
also known as the scale-by-scale scalar transfer budget equation (SBSS equation). In Eq. 2, Ac represents the advection ofδ c2 by the mean flow, Πc is the non-linear transfer term, ΠU,cis the linear transfer terms, Pc is the turbulent production, Tc isturbulent transport terms, Dκ is molecular diffusion in r space,
DX,κ is the molecular diffusion along X, and χ is scalar dissipation, At,c is time dependence terms. While there are manystudies on energy transfer, analysis for the scalar transfer is relatively limited. Therefore, the main object of this paper is set
to investigate the transport mechanism for scalar in the regular grid-generated turbulence with different Reynolds numbers
and at streamwise positions.
It has been widely used to investigate the mechanism invarious flows (Yasuda & Vassilicos, 2018; Zhou et al., 2020).
A similar discussion can be made for scalar by extending theequation proposed by (Hill, 2002), although analysis for thescalar transfer is relatively rare. Therefore, the main objectiveof the present research was set to investigate the inter-scaletransport mechanism for scalar in grid-generated turbulence.

Crank–Nicolson method is applied for time integration. The
periodic condition is employed in the vertical (y) and spanwise (z) direction for velocity and scalar. The advective outflow condition for the velocity and scalar field is adopted at
the outlet. The inlet condition of velocity is a uniform streamof U0 = 1.0, while the scalar is set to two parallel streams of
C0 = 0.0 and C1 = 1.0.

RESULTS AND DISCUSSION
Figures 2-4 show the distributions of the terms in the
SBSS equation for different ReM , respectively. Recent Studies (Obligado & Vassilicos (2019); Meldi & Vassilicos (2021))
mentioned that the tendency towards equilibrium for interscale energy transfer presents only in the vincity of the Taylor micro scale λ , and there is a systematic departure fromequilibrium as the scale r are considered further away from
λ . Since the form of the SBSS equation is similar to KHMH
equation, the equilibrium and non-equilibrium characteristicsof the inter-scale scalar transport may have similar behavior.
Hence, the Taylor microscale λ is also employed when normalizing the relative position r between two-points x and x′ inx coordinate. On the other hand, the y coordinate represents
the terms normalized by χ a , where the upper index a indicates the circumferential average. As illustrated by Figs. 2-4,
the molecular diffusion Daκ balances the dissipation χ a at thescale close to r = 0 regardless of the streamwise position andthe ReM , while other terms are almost zero. This is similar to
the energy transfer, in which conversion from kinetic energy
to thermal energy plays the main role when the scale is small.
Additionally, since there is a mean scalar gradient in the vertical direction, the production term Pca is large for large r. Meanwhile, the non-linear transfer term Πca reaches a peak aroundr/λ = 1 in all cases. Negative value of Πca , which indicates aninverse cascade of the non-linear scalar transport, is observedin some cases, while others do not show such behavior.

NUMERICAL METHODSIn this study, direct numerical simulations (DNSs) are performed. Figure 1 shows the schematic view of the computational domain. It is a rectangular box with Lx × Ly × Lz =32M × 6M × 6M, where M is the distance between two gridbars. The inlet Reynolds number based on the inlet velocity
U0 and the length of grid bar M, ReM , is set to 5, 000, 9, 000,
and 15, 000. The resolution calculated by ∆x/η does not exceed 2, where ∆x is the computational mesh size and η is the
Kolmogorov scale. The statistical results are collected over
100, 000 time steps with an interval of ∆t = 3.5 × 10−4 after
the flow reach the steady state. The initial courant number is
set to 0.3.
The governing equations are the dimensionless NavierStokes equation, the equation of continuity and the scalartransport equation. The Schmidt number Sc is set to 1. The
flow field is solved using the finite difference method with thefractional step method. The Poisson equation is solved by the
conjugate gradient (CG) method. The explicit/implicit hybrid
scheme based on the third-order Runge–Kutta method and the212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

x/M = 10

0

1

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

2

3

4

5

6

7

8
(c)

x/M = 20

0

1

2

3
r/λ

4

(b)

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-Πac/χa-ΠaU,c/χaΡac/χaΤac/χa
0

1

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

5

1
0.8
0.6
0.4
0.2
0
-0.2

-Αac/χax/M = 15

2

3

4

5

6
(d)

x/M = 25

0

1

2

r/λ

Daκ/χa
DaX,κ/χaΑat,c/χa

3

4

1
0.8
0.6
0.4
0.2
0
-0.2

5

Figure 2. Circumferentially averaged terms in SBSS equation normalized by the averaged scalar dissipation rate χ a at(a)x/M = 10, (b)x/M = 15, (c)x/M = 20, and (d)x/M = 25 incase of ReM = 5, 000.
(a)

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

x/M = 10

0

1

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

2

3

4

5

6

7

8
(c)

x/M = 20

0

1

2

3
r/λ

4

(a)

0
1.2
1
0.8
0.6
0.4
0.2
0
-0.2

1

2

3

4

5

6

7

x/M = 20

0

1

2

3
r/λ

4

5

8
(c)

ReM = 15000-Πac/χa at x/M = 10
-Πac/χa at x/M = 25

-Πac/χathat the high turbulent intensity restrains the process of scalartransport from small scale to large scale. Furthermore, the difference of Πca between the developing region and developed
region become more significant with the increase of the ReM .
For the sake of investigating the mechanism behind thisphenomenon, the third-order structure functions are introduced here to quantify the transport between different scales,
as shown in Eq. 3,-ΠaU,c/χaΡac/χaΤac/χa
0

1

2

3

4

5

6
(d)

x/M = 25

Daκ/χa
DaX,κ/χaΑat,c/χa

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5r/λx/M = 100 1 2 3 4 5 6 7 8
r/λ(c)Figure 5. The evolution of the normalized non-linear transfer
terms −Πc /χ a at different positions in cases of (a) ReM =5, 000, (b) ReM = 9, 000 and (c) ReM = 15, 000.

2⟨δ u∥ δ q2 ⟩ = ⟨(u∥ (x) − u∥ (x′ )) ui (x) − ui (x′ ) ⟩2⟨δ u∥ δ c2 ⟩ = ⟨(u∥ (x) − u∥ (x′ )) c (x) − c(x′ ) ⟩.

Figure 3. Circumferentially averaged terms in SBSS equation normalized by the averaged scalar dissipation rate χ a at(a)x/M = 10, (b)x/M = 15, (c)x/M = 20, and (d)x/M = 25 incase of ReM = 9, 000.

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

(b)

ReM = 9000

-Αac/χax/M = 15

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

5

0 1 2 3 4 5 6 7 8
r/λ

1
0.8
0.6
0.4
0.2
0
-0.2

0 1 2 3 4 5 6 7 8
r/λ

(b)

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

(a)

ReM = 5000

The third-order structure function for energy is also ploted for
a comapasion. Here the subscript || denotes the component of
the velocity vector parallel to the separation vector r. Also,a schematic view of the coordinates to express the function isshown in Fig. 6.

(b)

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

-Αac/χax/M = 15(3)

-Πac/χa-ΠaU,c/χaΡac/χaΤac/χa
0

1.2
1
0.8
0.6
0.4
0.2
0
-0.2

1

2

3

4

5

6
(d)

x/M = 25

0

1

2

r/λ

Daκ/χa
DaX,κ/χaΑat,c/χa3

4

5

Figure 4. Circumferentially averaged terms in SBSS equation normalized by the averaged scalar dissipation rate χ a at(a)x/M = 10, (b)x/M = 15, (c)x/M = 20, and (d)x/M = 25 incase of ReM = 15, 000.

To make this clearer, comparisons of Πca with different
ReM in the developing region (x/M = 10) and developed region (x/M = 25) were conducted. Figure 5 shows the results. The evolution of −Πca /χ a is subjected to ReM . With
the increase of ReM , −Πca /χ a tends to take a negative value
at relatively large r/λ regardless the streamwise positions, indicating that the minimum scale of the inverse cascade phenomenon becomes larger for larger ReM . This also implies

Figure 6. Schematic view of (a) the two points coordinate
system(b) the decomposition of the velocity vector.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
When energy or scalar transports from large scale to small
scale, the radial velocity difference δ u|| achieves a negativevalue. Thus, positive sign of −⟨δ u|| δ q2 ⟩ represents a normalcascade and it is the inverse cascade when its sign is negative. The distribution of circumferentially averaged third-order
structure functions over r for energy and scalar are shown in
Figs. 7 and 8, respectively. These figures reveal that boththe third-order structure functions for energy and scalar reacha peak at around r/λ = 1. However, the inverse cascade of
the scalar is observed for all cases, while almost only the normal cascade is observed for energy except the upstream region(x/M = 10). A possible reason causing this difference is thedifference of inflow conditions between velocity and scalar.
Therefore, a decomposition of Πc was conducted:

(a)

0.4
0.3
0.2
0.1
0
-0.1
-0.2

ReM = 5000

0

2

4

0.4
0.3
0.2
0.1
0
-0.1
-0.2

r/λ

6

8

(b)

0.4
0.3
0.2
0.1
0
-0.1
-0.2

10
(c)

ReM = 9000

0ReM = 15000

2

4

r/λ

6

8

10

-<δu||δc2>a/χar at x/M = 10-<δu||δc2>a/χar at x/M = 25

0

2

4

6

8

10

r/λ

−Πc = −∂
⟨δ u j δ c2 ⟩∂rjFigure 8. The evolution of the normalized third-order structure functions for the velocity at different positions in cases of(a) ReM = 5, 000, (b) ReM = 9, 000 and (c) ReM = 15, 000.

∂∂∂⟨δ uδ c2 ⟩ −⟨δ vδ c2 ⟩ −⟨δ wδ c2 ⟩
∂ rx
∂ rx
∂ rz
∂
(4)
−Πcx = −⟨δ uδ c2 ⟩∂ rx∂−Πcy = −⟨δ vδ c2 ⟩∂ ry∂−Πcz = −⟨δ wδ c2 ⟩
∂ rz
=−

0.8

λ

L

ReM = 5000

0.6

The expressions of the components of Πca in three directions
are shown in Eq. (4). Figure 9 displays the distributions at the
upstream region of x/M = 10 are displayed in Fig. 9. Boththe x and z components gradually decrease for r > λ and eventually converge to 0, while the y component decreases rapidly
and takes negative values after reaching the peak. In otherwords, it is found that Πcy causes the inverse cascade pheonmenon.

0.8
0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2

0.8

0 1 2 3 4 5 6 7 8
λL

ReM = 15000

0.6
0.4

λ

L

ReM = 9000

0 1 2 3 4 5 6 7 8r/λ-Πac/χa-Πacx/χa0.2-Πacy/χa0-Πacz/χa-0.2
0 1 2 3 4 5 6 7 8
r/λ

(a)

0.2
0.15
0.1
0.05
0
-0.05
-0.1

ReM = 5000

0

2

0.2
0.15
0.1
0.05
0
-0.05
-0.1

4

r/λ

6

8

10
(c)

ReM = 15000

Figure 9. The components of −Πac /χ a in three directions incases of (a) ReM = 5, 000, (b) ReM = 9, 000 and (c) ReM =15, 000.

(b)

0.2
0.15
0.1
0.05
0
-0.05
-0.1

ReM = 9000

0

2

4

r/λ

6

8

In order to further study this issue, we continue to decompose the Πc into two parts:10-<δu||δq2>a/εar at x/M = 10-<δu||δq2>a/εar at x/M = 25∂∂δv
∂ δ c2⟨δ vδ c2 ⟩ = −⟨δ c2
⟩−⟨
δ v⟩
∂ ry
∂ ry
∂ ry


∂ δ c2
1∂ v′ ∂ v− Πcy1 = −⟨δ v⟩ = − ⟨δ c2
+
⟩
∂ ry
2
∂ y′ ∂ y
 ′

∂δv
∂c
∂c
⟩ = −⟨δ vδ c+⟩
− Πcy2 = −⟨δ c2∂ ry∂ y′ ∂ y− Πcy = −

0

2

4

6

8

10

r/λ

Figure 7. The evolution of the normalized third-order structure functions for the velocity at different positions in cases of(a) ReM = 5, 000, (b) ReM = 9, 000 and (c) ReM = 15, 000.

(5)Thus, we further decomposed Πcy as shown in Eq. 5. Figa /χ a at x/M = 10 in caseure 10 shows the components of −Πcya gradually becomesof ReM = 5, 000. It illustrates that −Πcy1aconstant while −Πcy2 remains relatively large and maintains
a . Therefore, it can be conthe same evolution trend as −Πcycluded that the vertical flux of the partial derivative of scalara = ⟨ ∂ δ c2 /∂ r δ va ⟩ induces the inverse casvariance −Πcy2
ycade phenomenon for scalar at large scales.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
λ

0.4

L
-Πacy/χa-Πacy1/χa-Πacy2/χa

0.3
0.2
0.1

ReM = 5000 at x/M = 10

0
-0.1
-0.2

0

1

2

3

4
r/λ

5

6

7

8

Figure 10. The components of −Πacy /χ a in case of ReM =5, 000 at x/M = 10.

CONCLUSIONS
An investigation of the inter-scale scalar transport is conducted based on the DNS data of grid-generated turbulence.
The results reveal that the y component of the non-linear transfer term Πc , which is identical to the direction of the meanscalar gradient, induces the inverse cascade phenomenon forscalar when the scale is large.

ACKNOWLEDGEMENT
The numerical simulations were conducted at Institute of
Fluid Science, Tohoku University and and the Earth Simulator
Center (JAMSTEC). Part of this research was supported by
KAKENHI No.20K04264.

REFERENCES
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Meldi, M. and Vassilicos, J.C., 2021, “Structure-function
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Zhou, Y., Nagata, K., Sakai, Y., Watanabe, T., Ito, Y. andHayase, T., 2020, “Energy transfer in turbulent flows behind
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