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

DNS AND LES STUDIES ON TURBULENT HEAT TRANSFER
PHENOMENA OF ROUND IMPINGING JET IN FINITE VESSEL
H. Hattori
Advanced Manufacturing Research Center, Nagoya Institute of Technology
Gokiso-cho, Showa-ku, Nagoya 466-8555, Japanhattori@nitech.ac.jp
T. Mizukami, H. Baba, T. Houra and M. Tagawa
Department of Electrical and Mechanical Engineering, Nagoya Institute of Technology

INTRODUCTIONDNS AND LES OF ROUND IMPINGING JET
WITH HEAT TRANSFER

Technology of enhancement of heat transfer using turbulence is very important factor for increase of heat efﬁciencyof thermal equipment. In such a technology, since the impingement jet gives higher heat transfer rate around the stagnation point, the impinging jet is widely used for the coolingof wall, etc. Heat transfer rate, however, rapidly decreases
toward downstream region on the impingement wall, so it is
difﬁcult to effectively use the impinging jet for the enhancement of heat transfer in wide region. Thus, it is important
to know the characteristics, structures, and statistical quantities of turbulent heat transfer phenomena of impinging jet for
the enhancement of heat transfer rate using it. In the previous
study (Hattori & Nagano, 2004), the plane impinging jet withshort impingement distance has been investigated by direct numerical simulation (DNS), in which it is found the appearanceof the second peak of local Nusselt number. So, to enhanceheat transfer rate using the impinging jet, a short impingement
distance is very efﬁcient. On the other hand, many turbulent
heat transfer phenomena of the impinging jet in a ﬁnite vessel
which has both the impingement and side walls can be practically observed, where a round impingement jet is almost employed in. In this case, heat transfer rate on both impingement
and side walls should be investigated to know heat efﬁciency
in a whole ﬁnite vessel. In the practical engineering problems,
a long impinging distance is observed in many cases. Thus,
observations of heat transfer phenomena of impinging jet withlong impinging distance should be also required, but the DNS
is difﬁcult to use for this problems due to the calculation cost.
In this case, since a large eddy simulation (LES) and Reynolds
averaged Navier-Stokes simulation (RANS) which might givethe high accurate prediction are an efﬁciently technique for theobservations, the prediction accuracy of LES or RANS shouldbe evaluated using proper DNS or experimental data. Evenout of these simulations, the LES can be expected to be able toobserve the similar phenomena DNS obtained, but the proper
turbulence model should be employed in the LES.

The governing equations of cylindrical coordinates system for the DNS of round impinging jet with heat transfer
are the Navier-Stokes, the continuity and the energy equations
which are assumed as incompressibility ﬂuid as indicated below (Hattori et al., 2015).
1 ∂ (rur ) 1 ∂ uϕ ∂ uz
+
+
=0
r ∂r
r ∂ϕ
∂z

(1)

u2ϕ
1 ∂
∂ ur 1 ∂∂+(rur ur ) +(uϕ ur ) + (uz ur ) −
∂t
r ∂r
r ∂ϕ
∂z
r
[
(
)
1 1 ∂
ur
∂p
∂ ur
1 ∂ 2 ur ∂ 2 ur+r
− 2+ 2=−+∂ r ReD r ∂ r
∂r
r
r ∂ϕ2
∂ z2
]
2 ∂ uϕ
(2)
− 2
r ∂ϕ
ur uϕ
∂ uϕ 1 ∂1 ∂∂+(rur uϕ ) +(uϕ uϕ ) + (uz uϕ ) +
∂t
r ∂r
r ∂ϕ
∂z
r
[
(
)
2
uϕ
∂ uϕ
1 ∂p
1 ∂
1
1 ∂ uϕr− 2 + 2=−+r ∂ ϕ ReD r ∂ r
∂r
r
r ∂ϕ2
]
∂ 2 uϕ2 ∂ ur(3)+ 2 + 2∂zr ∂ϕ
1 ∂∂ uz 1 ∂∂+(rur uz ) +(uϕ uz ) + (uz uz )
∂t
r ∂r
r ∂ϕ
∂z
[
(
)
]
∂p
1 1 ∂
∂ uz
1 ∂ 2 uz ∂ 2 uz
+
r
+ 2
=−
+
∂ z ReD r ∂ r
∂r
r ∂ϕ2
∂ z2
∂θ 1 ∂∂
1 ∂+(rur θ ) +(uϕ θ ) + (uz θ )
∂t
r ∂r
r ∂ϕ
∂z
[
(
)
]
1 ∂
1 ∂ 2θ ∂ 2θ
1
∂θ
=
r
+ 2
+
ReD Pr r ∂ r
∂r
r ∂ϕ2
∂ z2

(4)

(5)

On the other hand, the governing equations for LES aregiven as follows:

In this study, the DNS of turbulent heat transfer phenomena of round impinging jet in a ﬁnite vessel which is a circular
container is conducted in order to observe and investigate characteristics of such heat transfer phenomena. Also, the LES of
turbulent heat transfer phenomena of round impinging jet in aﬁnite vessel is carried out, and the prediction of LES is evaluated using DNS data we obtained. DNS and LES are carriedout under conditions of isothermal wall and various impinging
distances to reveal an effect of impingement distance for an understanding of heat transfer phenomena, in which the detailed
turbulent statistics, structures and mechanism of heat transfer
are investigated.

1 ∂ (rūr ) 1 ∂ ūϕ ∂ ūz
+
+
=0
r ∂r
r ∂ϕ
∂z

(6)

ūϕ2
∂ ūr 1 ∂1 ∂∂+(rūr ūr ) +(ūϕ ūr ) + (ūz ūr ) −
∂t
r ∂r
r ∂ϕ
∂z
r
[
(
)
∂ p̄
1 1 ∂
∂ ūr
ūr
1 ∂ 2 ūr ∂ 2 ūr=−++r− 2+ 2∂ r ReD r ∂ r
∂r
r
r ∂ϕ2
∂ z2
] [
)
1 ∂
2 ∂ ūϕ1 ∂ (−− 2(rur ur − rūr ūr ) +uϕ ur − ūϕ ūr
r ∂r
r ∂ϕ
r ∂ϕ
]
)
1(
∂
(7)
+ (uz ur − ūz ūr ) − uϕ uϕ − ūϕ ūϕ∂zr

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
∂ ūϕ 1 ∂
ur uϕ
1 ∂
∂
+
(rūr ūϕ ) +(ūϕ ūϕ ) + (ūz ūϕ ) +
∂t
r ∂r
r ∂ϕ
∂z
r
[
(
)
∂ ūϕ
ūϕ
1 ∂ p̄
1
1 ∂
1 ∂ 2 ūϕ+r
− 2 + 2=−r ∂ ϕ ReD r ∂ r
∂r
r
r ∂ϕ2
] [
)
∂ 2 ūϕ2 ∂ ūr1 ∂ (+ 2 + 2−
rur uϕ − rūr ūϕ
r ∂r
∂z
r ∂ϕ
) ∂ (
)
1 ∂ (
uϕ uϕ − ūϕ ūϕ +uz uϕ − ūz ūϕ
r ∂ϕ
∂z
]
)
1(
− ur uϕ − ūr ūϕr∂ ūz 1 ∂1 ∂∂+(rūr ūz ) +(ūϕ ūz ) + (ūz ūz )
∂t
r ∂r
r ∂ϕ
∂z
[
(
)
]
∂ p̄
1 1 ∂
∂ ūz
1 ∂ 2 ūz ∂ 2 ūz
+
r
+ 2
+
=−
∂ z ReD r ∂ r
∂r
r ∂ϕ2
∂ z2
[
)
(
1 ∂
1 ∂
−
(rur uz − rūr ūz ) +uϕ uz − ūϕ ūz
r ∂r
r ∂ϕ
]
∂
+ (uz uz − ūz ūz )∂z

to close the equations of Eqs. (6)∼(10), the mixed time scalemodels for the eddy diffusivities in both the velocity and thermal ﬁelds (Inagaki et al., 2005, 2012) are employed. The DNS
and LES is carried out using the well-established high accuracy ﬁnite difference method with the cylindrical coordinate
system (Hattori et al., 2015, 2016), but the 6th-order accurate central difference is used for the spatial difference in ϕdirection in DNS and LES of this study. Also, the 3rd order
low storage Runge-Kutta method is introduced for the time advancement of all terms except for molecular diffusion terms.
The conﬁguration of calculation domain for a round impinging jet in a ﬁnite vessel is shown in Fig. 1, in which the
fully-developing turbulent pipe ﬂow with uniform temperature
produced in the driver part is set as the inlet condition, i.e.,the driver part gives turbulent jet. As for the outlet condition,a circular slit whose area is 13π D2 is arranged on the rim of
upper wall as indicated in Fig. 1. Turbulent jet impinges thelower (impingement) wall, and then the ﬂuid ﬂow ﬂows along
the impingement wall and side wall, and a part of ﬂow drainsat the slit. Also, the impinging jet with turbulence yields heattransfer phenomena due to the isothermal wall condition on
both the lower and side walls, where the upper wall is set as theadiabatic wall. The impingement distances are set H/D = 2, 4
and 8. In the cylindrical coordinate system, since the grid resolution of radial direction becomes large toward downstream
region on the impingement wall, the special attention is paid
for the arrangement of girds (for instance, the grid resolutionsof r+ ∆ϕ are 0.02 ∼ 2.80 in the DNS case of H = 4D, and thetwice resolution is used for the LES). The grid points are arranged as 192 points for both the radial and circumferential
directions in the DNS, and 96 points for both the radial andcircumferential directions in the LES. The grid points for the
impinging wall normal direction are arranged as 192 (case of
H/D = 2) , 384 (H/D = 4) and 768 (H/D = 8) for the DNS,
and 96 (case of H/D = 2) , 192 (H/D = 4) and 384 (H/D = 8)
for the LES. The nondimensional time step, ∆t, is 7.2 × 10−4 ,and the DNS and LES are conducted until the well-developed
turbulent statistics are obtained.
On the other hand, in order to preserve the energy balance in the calculation domain, the following relation should
be maintained.

+∂ θ̄ 1 ∂∂1 ∂+(rūr θ̄ ) +(ūϕ θ̄ ) + (ūz θ̄ )
∂t
r ∂r
r ∂ϕ
∂z
]
[
(
)
∂ θ̄
1
1 ∂
1 ∂ 2 θ̄ ∂ 2 θ̄
+ 2
=
r
+ 2
ReD Pr r ∂ r
∂r
r ∂ϕ2
∂z
[
) 1 ∂ ()
1 ∂ (uϕ θ − ūϕ θ̄−
rur θ − rūr θ̄ +
r ∂r
r ∂ϕ
]
)
∂ (
uz θ − ūz θ̄z
+
∂z

(8)

(9)

(10)

where (¯) means the ﬁltered value for LES. The coordinates r,
ϕ , z are the coordinate in the radial, circumferential and impingement wall-normal directions, respectively. Also, ur , uϕ ,uz are the velocity components of each direction, respectively.
Also, t is time, p is pressure, ν is the kinematic viscosity andα is the thermal diffusivity. In the DNS and LES, equations
are nondimensionalized by the bulk velocity at the inlet, Uzm ,the radius of inlet pipe, R (= D/2), where D is the diameter of inlet pipe, and temperature difference between the inlet
and impingement wall temperature, ∆Θ (= Θin − Θw ). As forthe calculation parameters appeared by the nondimensionalized equations, the Reynolds number based on the bulk velocity and the diameter of inlet pipe, ReD (= (Uzm D)/ν ), is set
as 4300, and the Prandtl number, Pr (= ν /α ), is given as 0.71
assuming the air.
As for the turbulence models of LES which are required[]
Ein · π R2 + qw π (14R)2 + 2π · 14R · H[]=Eout π (14R)2 + π (12R)2

(11)

(
)
where, Ein = ρ c pUzm Θin is the energy from the driver pipe,
qw (= λ (∂ θ /∂ z)|lower − λ (∂ θ /∂ r)|side ) (is the heat ﬂux from)both the lower and side walls, and Eout = ρ c pUout Θout is

Nu=(ReDPr)102

2
ReD = 4300
Pr = 0.71

Magagnato et al. (2021)
IFW ReD = 5300 H/D = 2

1

Solid lines: Modified
Dashed lines: Not modified

00
Figure 1. Computational domain, coordinate system andwall thermal conditions

Figure 2.

2

H/D = 2
H/D = 4
H/D = 8

2

4

r/D 6

Distributions of modiﬁed local Nusselt numbers

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
the energy at the rim. Although this relation generally may
be maintained by the boundary conditions, the inadequate results can be observed in the calculations. Thus, this relationis adopted so as to correct the energy balance in the calculations. Figure 2 shows the comparison between the results oflocal Nusselt number with correction and without correction.
Obviously, the effect of correction can be observed, and theresult is close to the other DNS result (Magagnato et al., 2020)in the downstream region.

the large recirculation region in the downstream region due toexisting the top and side walls and the separation region near
the side wall are observed in all cases.
The vortex structures detected by the second invariant ofgradients of velocity is indicated in Figs. 4. The vortex structures can be seen from the inlet, because the fully developed
turbulent pipe ﬂow blows from the inlet, and the structures
are also found on the impinging wall. On the impinging wall,the vortex structures expand toward downstream region in allcases. In the case of H/D = 2 as shown in Fig. 3(a), the vortex
structure can be seen on the side wall. In cases of H/D = 4
and 8, the vortex structures are not detected in the downstream
region. This is because the laminarization of ﬂow appears due
to decrease of the ﬂow rate of cross-sectional area which becomes large in the downstream region.

RESULTS AND DISCUSSION
Flow ﬁeld structures

z/D

Figures 3 show the streamline of round turbulent impinging jetin the ﬁnite vessel obtained by DNS. In the turbulent jet blown from the inlet pipe, the potential core region canbe seen in all cases, and then it is found the region vanishednear the impinging wall due to the effect of wall in cases of
H/D = 2 and 4. In the case of H/D = 8 as shown in Fig. 2(c),the potential core region of round jet is difﬁcult to observe
lower than z/D = 4, and then the velocity is decreased by both
effects of development of round jet and the wall. It can beobserved the ﬂow impinges the wall, and then the ﬂow redeveloped along the impingement wall. Moreover, appearance of

2

1

0

1

2

3

4

5 r/D 67(a) H/D = 2(a) H/D = 2

z/D

4

2

0

1

2

3

4

5 r/D 67(b) H/D = 48(b) H/D = 4
z/D

6

4

2

0

1

2

3

4

5 r/D 67(c) H/D = 8(c) H/D = 8

Figure 3. Streamline in round turbulent impinging jet withheat transfer by DNS

Figure 4. Vortex structure in round turbulent impinging jetwith heat transfer by DNS

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Fundamental turbulent statistics and evaluation of LES predictions

side wall. Thus, there is the inﬂuence of side wall near the sidewall in this diameter of vessel, but the effect of side wall does
not almost affect around the stagnation point.
On the other hand, it is well-known that the stagnation
point values of Nu are proportional to the impinging distancein general, but the stagnation point values of Nu are not so as
shown in Fig. 5(b), in which the almost similar values of local
Nusselt number in the cases of H/D = 2 and 4 are obtained.
The reason why these local Nusselt numbers are obtained is
discussed in the next section. In the downstream region fromthe stagnation point whose local Nusselt number takes the peakvalue, local Nusselt number slowly decreases on the impinging wall toward the side wall, and it can be also observed thatthe similar values of local Nusselt number in the all cases inthe downstream region (r/D > 5). Near the side wall, the local Nusselt number of all cases slightly increases near the sidewall due to both the separation and the inﬂuence of side wallwhose thermal boundary condition is also a constant wall temperature.
The predicted local wall friction coefﬁcients and local
Nusselt numbers almost agree with the DNS results in allcases, but slight discrepancies of predicted Nusselt number
can be observed near the point whose local friction coefﬁcienttakes the largest value in the region, i.e., overpredictions of local Nusselt number can be observed. Near the region, since
the SGS eddy diffusivity for momentum becomes large, the
SGS eddy diffusivity for momentum remarkably affects the
SGS eddy diffusivity for heat. Thus, overpredictions of local Nusselt number are obtained. As for this prediction, theLES using the standard Smagorinsky SGS model (Smagorinsky, 1963) with the wall damping and a constant SGS Prandtl
number function is also carried out, but the similar results are
obtained.
In order to observe the heat transfer performance, the ratio of wall friction coefﬁcient and j-factor which is deﬁned
by jH = Nu/(ReD Pr1/3 ), which indicates the efﬁcient of heattransfer and the similarity between the velocity and thermalﬁelds as shown in Fig.6. Since the ratios are over unity between r/D = 1 and 3, the inﬂuence of friction force of ﬂow
is greater than that of heat transfer in this region. And, near
the stagnation point and side wall, the values in all cases takeunder unity, so the efﬁcient of heat transfer is larger than thefriction force, i.e., the heat transfer performance is well. Near
the stagnation point, the reason why the heat transfer performance is well the thermal boundary layer faster develops thanvelocity boundary layer. In the downstream region near theside wall, it can be considered that the heat transfer performance is well due to the laminarization and separation. Also,from this result, the values of ratio obviously indicate the dissimilarity between velocity and thermal ﬁelds in all cases.

As a fundamental turbulent statistics, local wall friction
coefﬁcients and local Nusselt numbers along the impinging
wall are indicated in Fig. 4. Note that results of LES are included in the following ﬁgures for the evaluation of LES predictions. The distributions of local friction coefﬁcient obtainedby DNS on the impinging wall indicate a rapid increase, andthen take the peak value given away from the stagnation point
in all cases. The peak values are obtained proportional to theimpinging distance, in which the shortest impinging distancegives the largest peak value as shown in Fig. 4(a) . From thepoint taking peak value, the friction coefﬁcients of all cases
slowly decrease toward downstream region, and ﬁnally thesevalues become similar independently of impinging distance inthe downstream region (r/D > 5). Also, near the side wall,
the separation where the friction coefﬁcient becomes negative value can be seen due to both the laminarization whichis caused by becoming large cross section and the effect of

6
ReD = 4300
Pr = 0.71

Cf102

4DNS LES H=D
2
4
8

2
0
0

2

4

r/D

6

(a) Local wall friction coefﬁcients
50
ReD = 4300
Pr = 0.71

Nu

DNS LES H=D
2
4
8

25

0
0

2

4

r/D 6

(b) Local Nusselt numbers
Figure 5. Distributions and comparisons of LES predictions
with DNS: Fundamental parameters(Cf=2)=jH

4

Distributions of turbulent quantities and evaluation of LES predictions

DNS LES H=D
2
4
8

3
2

The distributions of radial wise mean velocities near thestagnation point and the wall are shown in Fig. 7, in which thepredictions of LES are included for the evaluation. Since theradial wise mean velocity develops as the wall jet along theimpinging wall, two normalizations are adopted. Figure 7(a)shows the radial wise mean velocities normalized by the characteristic velocity, Uzm , and the mean velocities normalized byboth the maximum value of local mean velocity, Umax and thehalf-value width of the wall jet in the free shear region, z1/2 .
The proﬁles of radial wise mean velocity obviously indicate
similar proﬁle without regard to the impingement distance asshown in Fig. 7(b), though the different proﬁles are shown in

10

ReD = 4300
Pr = 0.71

-1
0

2

4

r=D 6

Figure 6. Distributions and comparisons of LES predictions
with DNS: Ratio of wall friction coefﬁcient and j-factor

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Fig 7(a) . As for the LES predictions, it can be found that theradial wise mean velocity are properly predicted by the LES.
Figures 8 indicate the distributions of Reynolds shear

Ur=Uzm1
z=D

DNS LES H=D
2
4
8

0.5

1

z/D

ReD=4300
Pr=0.71

0.5

0

ÇÝ0.3

0.2

DNS LES H=D
2
4
0.1
8

r=D

(a) Normalized by the characteristic velocity

00

Ur=Umax3

ReD=4300
Pr=0.71

0.5

r/D

1

(a) r/D = 0 ∼ r/D = 0.5

DNS LES H=D2
48

Ç=ÐÇÝ0.3

0.2
ReD = 4300
Pr = 0.71
DNS LES H=D
2
4
0.1
8

z/D

z=z1/22

1

0

ReD = 4300
Pr = 0.71

0.5

1

r=D

(b) Normalized by the local mean velocity at the halfvalue of width0

DNS LES H=D
2
4
8

2

{uzíÝ50

0.2

ReD = 4300
Pr = 0.71

z/D

z/D

r/D

Figure 9. Distributions and comparisons of LES predictions
with DNS: Mean temperature2
Ý30{uruz=Uzm

ReD = 4300
Pr = 0.71

1.5(b) r/D = 1.0 ∼ r/D = 1.5

Figure 7. Distributions and comparisons of LES predictions
with DNS: Radial wise mean velocity

2

1

1

DNS LES H=D
2
4
8

0.1

0

0.5

1

r/D

0

(a) Normalized by the characteristic velocity

0.5

r/D

1

(a) r/D = 0 ∼ r/D = 0.5

2{uruz=Umax
Ý30

3ReD = 4300
Pr = 0.71{uzí=ÐÇUzmÝ30

0.2

DNS LES H=D248

z=z1/2z/D

2ReD = 4300
Pr = 0.71

0.1

1 DNS LES H=D
0

0

2
4
8

0.5

1

r/D

0

(b) Normalized by the local mean velocity at the halfvalue of width

1

1.5

r/D

2

(b) r/D = 1.0 ∼ r/D = 1.5

Figure 8. Distributions and comparisons of LES predictions
with DNS: Reynolds shear stresses

Figure 10. Distributions and comparisons of LES predictions
with DNS: Wall-normal turbulent heat ﬂuxes

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022stresses in wall-normal direction with z/D and z1/2 , in which
the predictions of Reynolds shear stress by LES are also included in these ﬁgures, and it can be seen that the predictions
are in good agreement with DNS results. In Fig. 8(a), the proﬁle of Reynolds shear stress in the case of H/D = 2 can bedifferently seen in comparison with other cases, but the similar proﬁles can be found in Fig. 8(b). Thus, adopting the halfvalue width of the wall jet in the free shear region, z1/2 , thenear-wall distribution of Reynolds shear stress can be similarly demonstrated with the different impinging distance. The
proﬁles, however, do not indicate similar proﬁle as like the proﬁles of radial wise mean velocity, because the points of nearwall maximum values of Reynolds shear stress in each casedisagree. This discrepancies of Reynolds shear stress distributions give the different peak values of local wall friction coefﬁcient. Also, it can be found in the normalization using z1/2
that the points which are the zero radial wise mean velocity
gradient disagree with the points which are zero of Reynolds
shear stress.
Regarding the thermal ﬁeld, the proﬁle of mean temperature distribution forms as a boundary layer, so the normalization of wall-normal distance using z1/2 may not be adopted.
Thus, distributions of mean temperature from the stagnation
point with the wall-normal distance, z/D, are shown in Fig. 9,
in which the LES predictions are also included. It can be foundthat the LES predictions give disagreement results in comparison with DNS results. As for the DNS results, it can beobserved that the temperature gradient near the wall becomes
small in all cases toward the downstream region, which is consistent with the distributions of local Nusselt number. This isbecause the local Nusselt number only depends on the temperature gradient due to a constant temperature condition on theimpinging wall.
Figure 10 shows the distributions of wall-normal turbulent heat ﬂux including the LES predictions. At the stagnation
point, the wall-normal turbulent heat ﬂux of LES predictionsin the cases of H/D = 2 and 4 gives overpredictions near thewall in comparison with DNS results. Since the temperature
gradients of these cases almost agree with the DNS result, it
can be seen that these overpredictions are difﬁcult to affect thepredicted proﬁles of mean temperature. In view of DNS result,
the wall-normal turbulent heat ﬂux does not inﬂuence the temperature proﬁle in the case of H/D = 2, but the wall-normal
turbulent heat ﬂux slightly affects the temperature proﬁle in thecase of H/D = 4. Thus, obtaining the similar values of local
Nusselt number at the stagnation point in the cases of H/D = 2
and 4 as indicated in Fig. 5(b) is due to the difference effect ofwall-normal turbulent heat ﬂux, i.e., turbulence does not affectat the stagnation point in the case of H/D = 2. Returning theevaluation of LES prediction, observing the rapid increases of
local Nusselt number in all cases around the point taking maximum value of friction coefﬁcient as shown in Fig. 5(b) is due todiscrepancies of wall-normal turbulent heat ﬂux between LES
predictions and DNS results as indicated in Fig. 10(b). As forthe this reason, the effect of SGS eddy diffusivities remarkably
inﬂuence as mentioned forgoing section, so it may be necessary to pay attention when an LES of round impinging jet withheat transfer using Smagorinsky SGS type turbulence model iscarried out.

an LES prediction and to observe the heat transfer phenomena in a ﬁnite vessel. The predictions of LES are evaluatedusing DNS data, as the result, it is assessed that the LES canqualitatively predict such phenomena. In the results of DNS,
the effect of impingement distance for the heat transfer on theimpinging wall can be clearly found, but similar local Nusselt numbers at the stagnation point are obtained in both thecases of H/D = 2 and H/D = 4. Concerning the reason for
obtaining similar values, it is found that the distributions oftemperature and the impinging wall normal turbulent heat ﬂux
near the wall are related, in which the wall normal turbulent
heat ﬂux remarkably affects in the case of H/D = 4. Also,
the streamline and vortex structure are obviously indicated toshow the characteristics of ﬂow situation in the ﬁnite vessel. In
the downstream region on the impinging wall, since the laminarization appears due to becoming large cross section andthere is the side wall, it can be seen that the ﬂow separation
occurs in all cases. The effect of side wall, however, only
inﬂuences near the side wall, but the effect can not be seenaround the stagnation point in this diameter of vessel. As forthe efﬁciency of heat transfer, the wall frictions in all cases are
greater than heat transfer rate around the point taking the maximum value of friction coefﬁcient. Thus, it is found that the effective heat transfer rate is mainly obtained around stagnation
point. Therefore, DNS and LES reveals the characteristics,
structures, and statistical quantities of turbulent heat transfer
of a round impinging jet with various impinging distances in aﬁnite vessel.

ACKNOWLEDGEMENTS
This work is the result of a collaborative research programwith the Research association of Automotive Internal Combustion Engines (AICE) for ﬁscal years 2020-2021.

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CONCLUSIONSThe DNS and LES of turbulent heat transfer phenomenaof round impinging jet in a ﬁnite vessel with various impinging distances are successfully carried out in order to evaluate

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