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

DISSIMILARITY BETWEEN THERMAL AND FLOW FIELDS IN TURBULENCE PIPEFLOW WITH CIRCUMFERENTIALLY-VARYING SURFACE HEAT FLUX

Yasuo Hattori
Sustainable System Research Laboratory
Central Research Inst. of Electric Power Industry
1646 Abiko, Abiko-shi, Chiba-ken, 270-1194, Japanyhattori@criepi.denken.or.jp

Yuma Hasebe
Numerical Analysis Group
Denryoku Computing Center, Ltd.
3-5-1 Kanda-nishiki, Tokyo, 101-0054, Japan
hasebe@dcc.co.jp

Hitoshi Suto
Sustainable System Research Laboratory
Central Research Inst. of Electric Power Industry
1646 Abiko, Abiko-shi, Chiba-ken, 270-1194, Japansuto@criepi.denken.or.jp

Keisuke Nakao
Sustainable System Research Laboratory
Central Research Inst. of Electric Power Industry
1646 Abiko, Abiko-shi, Chiba-ken, 270-1194, Japan
nakao@criepi.denken.or.jp

Shuji Ishihara
Numerical Analysis Group
Denryoku Computing Center, Ltd.
3-5-1 Kanda-nishiki, Tokyo, 101-0054, Japanishihara@dcc.co.jp

ABSTRACT
We performed well-resolved LESs for a turbulence pipe
flow under a circumferentially-varying surface heat flux. Afterconfirming the agreement of turbulence statistic with previous
DNSs through discussion on dependencies numericalparameters, such as a domain size, grid resolutions, SGS
models, we examined effects of the surface heat flux on
turbulence characteristics of temperature fields in the boundary
layer, especially paying attention to dissimilarities withvelocity fields. The surface heat flux distributions increased
temperature fluctuations and also activated intermittent
entrainment motions with high temperature fluids near theheating start points. The entrainment motions weakened
negative correlations between streamwise velocity withincreasing in characteristic length of temperature fields. Suchmodulation in large-scale structures of temperature fields must
be an origin of dissimilarities with velocity fields.

corrections to the model with a constant turbulent Prandtl
numbers are needed. This suggestion also implies thedissimilarity between thermal and flow fields, but the origin of
the dissimilarity has not been discussed yet.
In the present study, to clarify the origin of the dissimilarity,
we have carried out well-resolved LESs for a turbulence pipe
flow with circumferentially-varying heat flux. The Reynolds
numbers normalized with the pipe radius and the friction
velocities were set to 180 and 360, corresponding to theprevious DNSs (Antoranz et al., 2015). After confirming theagreement of turbulence statistic with previous DNSs throughdiscussion on dependencies numerical parameters, such as grid
resolutions, SGS models, we investigated changes inturbulence characteristics of temperature fields in the boundary
layer due to surface heat flux distributions and discussed
differences with velocity fields.
Reynolds number
dependencies also examined by using comparison among two
Reynolds number conditions.

INTTRODUCTION
Turbulent pipe flows under non-homogeneous heatconditions along the azimuthal direction are still unknown,
while turbulence characteristics of pipe flows under isothermal
conditions, including without surface heating, have been
widely investigated by using experiments and numericalsimulations; previous studies have mainly focused on ideal
thermal conditions, such as isothermal or uniform heat flux
conditions. Indeed, as far as the authors know, there exists
only one article dealing with a circumferentially-varying heat
flux condition (Antoranz et al., 2015), whereas such a conditionis one of the most simple configuration on non-homogeneous
thermal conditions and also is of practical interest inengineering applications relating to solar radiation, e.g. heatremoval design for solar power towers (Antoranz et al., 2015).
The article mainly discussed on the reproductively of
turbulence statistics with RANS models and suggested that

NUMERICAL SET-UP
The object we considered here was a turbulence flow,which was a pressure-driven incompressible flow of a fluidwith constant properties. The fluid temperature was simply
treated as a passive scaler: the buoyancy was not considered.
T h e t h e r m a l c o n d i t i o n i m p o s e d a t t he w a l l w a s acircumferentially-varying heat flux given by a sine function (qw= < qw >sin for 0 <  < , and qw = 0 for - <  < 0; where <qw > and  are the net heat flux at the wall and the azimuthal
direction, respectively), which also mimics heat flux conditionsof a pipe surface due to the sun radiation. The added heat leads
to a net increase of the temperature along the axial direction,and thus we set a heat balance with the axial gradient of thebulk temperature. Two Re number and Pr number conditions
were considered. The values of Re numbers normalized withthe pipe radius and the friction velocities were set to 180 and1

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

t'+

DNS

Figure 1. Grid arrangement.
Figure 2. Profiles of r.m.s. values of temperature fluctuations inboundary layer at  = /2 for Re=360, Pr =0.7, predicted byvarious grid resolutions. The DNS data by Antoranz et al.
(2015) are also plotted by squares.

360 for two Pr number conditions, Pr = 0.7 and 4,
corresponding to previous DNSs (Antoranz et al., 2015).
However, this paper will mainly discuss the results for Pr = 0.7.
The LESs were carried out with an open source CFD code,
OpenFOAM, which has been widely used all over the world fornumerical turbulence simulating. The present simulation was
based on a LES version with PIMPLEfoam of OpenFOAM, inwhich the option of RANS scheme was turn off. An
anisotropy-resolving subgrid-scale model proposed by Abe(2013), which improves the predictive performance particularly
for coarse grid resolution in the near-wall region, was adopted
to represent the subgrid-scale diffusion. The PrSGS was set at0.85. The differencing schemes for a limited-linear scheme foradvection terms and a back Euler scheme for time integration
were chosen.
The grid arrangement used in the present study is shown in
Fig. 1, which ensured the alignment of numerical grids withflow patterns, e.g., the grid spacing changed with the distancesfrom the pipe surface. The minimum grid level in the normal
direction with the wall unit was set to 0.57 and 1.1 for Re =180 and 360, respectively, corresponding to previous DNS(Antoranz et al., 2015). The computational domain had a longof 100R, where R is the radius of the pipe. The domain size isquite large compared with previous studies; this is because thatrecent studies (e.g. Guala et al., 2006; Monty et al. 2007; Lee etal., 2019) have suggested that the coherence structure, which
has a large characteristics length in the streamwise direction,
play in an important role in turbulence transport phenomena ofboundary layer flows.
The statistics were estimated by the Reynolds
decomposition for the time-series during 150  300, normalizedwith the wall units when there exists no increasing or
decreasing tendencies of the bulk parameters and fluctuation
components of velocities and temperature against time.
Firstly, we checked the dependencies of numericalparameters, such as the SGS model, grid resolutions, domain
size, on profiles of turbulence statistics. Figures 2 and 3 show
examples of r.m.s. values of temperature fluctuations, t’,predicted by various grid resolutions and SGS models. The
DNSs data (Antoranz et al. 2015) are also plotted in the figures.
The profiles in the boundary layer at  = /2, where the r.m.s.
value takes maximum value, are presented. The profiles arenormalized with the inner parameters, i.e., the friction velocity,u, the kinetic viscosity, and the friction temperature, t =<qw>/Cpu, where , Cp are density and constant pressure
specific heat of the working fluid, respectively. The normal

t'+

DNS

Figure 3. Profiles of r.m.s. values of temperature fluctuations inboundary layer at  = /2 for Re=180 and 360, Pr =0.7 and 4.0,
predicted by Smagrinsky model. The DNS data by Antoranz etal. (2015) are also plotted by squares.
grid resolution used in the present LESs gives good agreementwith DNSs, while LESs and DNSs are naturally in better
agreement with finer grid resolutions. Also, the profiles
predicted with Abe model agree well with those by DNSs,
whereas the profiles predicted by Smagrinsky model apparently
differ from those by DNSs, regardress to Re and Pr numbers.
Notice that, the profiles of r.m.s. values of temperaturefluctuations clearly showed the domain-size dependencies, i.e.,
the r.m.s. values near the wall with the domain size of 5R are
significantly larger than those with the domain size of 100R;
these overestimates of r.m.s values with the domain size of 5R
occur regardless to conditions, Re and Pr, which indicates thenecessity of large domain size. Moreover, we examined the
profiles of statistics in velocity fields under the homogeneousheat conditions and confirmed the profiles predicted by Abe
model with normal grid resolution, including the near-wallregion, agree well with those obtained DNSs reported byprevious studies (Fukagata and Kasagi, 2002; Piller, 2005),
including the Re number effects which give increases in r.m.s.
values and extensions of the logarithmic layer to outer regions.

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

120
/2 Re0360
180(constant heat flux, Re=360)

80

T+

/4

60

/4

/2 Re360
180(constant heat flux, Re=360)

0.8

S(t)

100

1.2
0

0.4

40
20

0.0

0
-20

100

101 +

y

-0.4

102

Figure 4. Profiles of time-averaged values of temperature inboundary layer at  = 0, /4, /2.

/4

/2 Re0360
180(constant heat flux, Re=360)

3.4

8

102

3.2

/4

/2 Re360
180(constant heat flux, Re=360)

3.0

6

2.8

4

2.6

2
0

y

3.6
0

F(t)

t'+

10

101 +

Figure 6. Profiles of skewness factors of temperature inboundary layer at  = 0, /4, /2.

14
12

100

100

101 +

y

2.4

102

100

101 +

y

102

Figure 5. Profiles of r.m.s. values of temperature in boundary
layer at  = 0, /4, /2.

Figure 7. Profiles of flatness factors of temperature in boundary
layer at  = 0, /4, /2.

RESULTS AND DISCUSSION
Profiles of time-averaged values of temperature, T, in theboundary layer at  = 0, /4, /2 are presented in Fig. 4. The
results under constant heat flux condition, the net heat flux of
which consistent with the present value, are also plotted in the
figure. The temperature in the boundary layer essentially
increases with , which is due to the circumferentially-varying
heat flux given by a sine function. The Reynolds number
dependencies seem to be small, while the temperature in theouter layer of boundary layer slightly decreases with Reynolds
numbers.
Figure 5 shows the profiles of r.m.s. values of temperature,t’, in the boundary layer at  = 0, /4, /2 with that for constantheat flux condition. The r.m.s. values drastically increase withcircumferentially-varying heat flux at the pipe surface. The
r.m.s values in the boundary layer are apparently large
compared with that for constant heat flux conditions, even at 
= 0, where the time averaged temperature is much low. The
maximum r.m.s. values in the boundary layer appear at y+ 30,
except at  = 0. Also, the r.m.s values, regardless to azimuthal
direction, naturally become constant approaching the center ofthe pipe. The Reynolds number dependencies seem to be small,while the r.m.s values in the outer layer of boundary layerslightly decreases with Reynolds numbers, except at  = 0.
The profiles of higher-order moments, skewness andflatness factors, S(t) and F(t), in the boundary layer arepresented in Figs. 6 and 7. The positive value of skew factor
with large flatness factor appear at  = 0, indicating theexistence of intermittent entrainment motions with hightemperature fluids. Such entrainment motions become more
active in the outer layer for Re = 180. Here, I must stress thatthese factors give no evidence for generation of coherence
structures in the near-wall region of thermal fields; the flatness
factors gradually increase with y,+ whereas the flatness factorsof streamwise velocities drastically increases in the near-wallregion.
Figure 8 shows the profiles of cross-correlation between
streamwise velocity and temperature in the boundary layer.
They are negatively correlated in the whole region. Negative
correlation in the boundary layer apparently weaken at  = 0
and also near the center of the pipe where the intermittent fluid
motions become active. The Reynolds effects only appear near
the center of the pipe, which rapidly decrease negative
correlations between streamwise velocity and temperature.
Figure 9 depicts a snapshot of streamwise velocity andtemperature in a cross section and a longitudinal section in thepipe. The flow and thermal fields in the cross section for  > 0,show the high-speed regions with low temperature flows evennear the wall with the organized patterns against the azimuthal
direction, corresponding to helically inclined structures (Ahnand Sung, 2017). Such structures in the thermal field graduallydiminish for  < 0, yielding the asymmetric patterns in the3

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

0.4

Rwt

0.2

0

/4

/2 Re

(a)

360
180

0.0

(b)

-0.2
-0.4

(c)

-0.6

Figure 10. Snapshot of contour of temperature fluctuations at
y+ = 3, (a), 30, (b), and 150, (c), normalizing with the innerparameters for Re = 360.

-0.8
-1.0

100

101 +

y

102

Figure 8. Profiles of cross-correlation between streamwise
velocity and temperature in boundary layer at  = 0, /4, /2.

(a)
(b)

(a)

(c)
Figure 11. Snapshot of contour of fluctuations of streamwise
velocity at y+ = 3, (a), 30, (b), and 150, (c), normalizing with
the inner parameters for Re = 360.

1.2

(b)

1.2

(a)

0

/4

/2 Re

1.0

360
180

0.8

Rtt

Rtt

1.0

0.6

0.4

0.2

0.2

5

10

15

z/R

-0.2
0

20

1.2
0

(c)

/4

/2 Re

360
180

0.6

1.0

Rww

Rtt

360
180

5

10

15

z/R

20

1.2

0.8

30

130 Re
360
180

0.6
0.4

0.2

0.2

0.0

3

(d)

0.8

0.4

-0.2
0

/2 Re

0.0

-0.2
0

1.0

/4

0.6

0.4

0.0

0

(b)

0.8

0.0
5

10

z/R

15

20

-0.2
0

5

10

z/R

15

20

Figure 12. Two point correlation in streamwise direction oftemperature at y+ =3, (a), 30, (b), 150, (c), and streamwise
velocity, (d).

Figure 9. Snapshot of contour of streamwise velocity, (a), andtemperature (b), normalizing with the inner parameters for Re
= 360.

of s/R > 0, where the heat flux at the surface is positive,
generates low- and high-temperature patterns elongated in thestreamwise direction.
Figure 12 shows two-point correlations of streamwise
velocities and temperatures fluctuations in the streamwise
direction at y+ = 3, 30, 150. These correlations clearly show
the differences of characteristics lengths in the streamwise
direction between the flow and the thermal fields. The
characteristics length of the flow fields gradually increases with
distances from the pipe surface. On the other hands, thecharacteristics length of the thermal fields strongly depends on
azimuthal positions as well as distances from the pipe surface,
and the longest characteristic length appears at  = 0 of nearwall region (y+ = 3). This length significantly large comparedwith the characteristics length of streamwise velocities.

cross and longitudinal sections.
Snapshots of fluctuations of temperature and streamwise
velocity at y+ = 3, 30 and 150 by a view with transformationfrom the true to the Cartesian (s-z) coordinates, where s = r,according to previous studies (Monty et al., 2007; Hellström
and Smiths, 2017) for Re = 360 are presented in Figs. 10 and
11. Normalization was done by the inner parameters. Lowand high-speed regions elongated in the streamwise direction
with meandering in the spanwise direction are generated in theflow fields, regardless to the azimuthal position. The length oflow- and high-speed regions in the streamwise direction
sometimes exceed 25R, while such structures seem to different
intensities and characteristic lengths against the distance fromthe pipe surface. Contrary to the flow fields, the thermal fields
significantly depend on the azimuthal position, while the region

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
These results indicate that the circumferentially-varying
heat flux at the pipe surface increased temperature fluctuations
and also activated intermittent entrainment motions with hightemperature fluids due to heterogeneous surface heat flux along
the azimuthal directions. The entrainment motions weakened
negative correlations between streamwise velocity withincreasing in characteristic length of temperature fields. Suchmodulation in large-scale structures of temperature fields must
be an origin of dissimilarities with velocity fields. Such largescale structures might be related to the helically included
structures in the pipe (Ahn and Sung, 2017). On the otherhands, the relationship with very large-scale motions (VLSMs),
which are energetic, typically containing half of the turbulencekinetic energy of the streamwise component, and with thecharacteristic length of 25R in the streamwise direction oflogarithmic layer (Guala et al., 2006; Monty et al., 2007;
Hellström and Smits, 2017; Lee et al., 2019) should be
carefully examined with helps of additional LESs for highReynolds number conditions. The Reynolds number of thepresent LESs are apparently small compared with that forexperiments to examine VLSMs.
Monty et al. (2007)
experimentally investigated the features of VLSMs of
turbulence pipe flows with the range of Re from 615 to 4355
and confirmed the existence of VLSMs regardless to Reynolds
number. We have a plan to carry out the LESs at Re = 615.

Fukagata, K., and Kasagi, N., 2002, “Highly energyconservative finite difference method for the cylindricalcoordinate system,” J Comp Phys, Vol.181, pp. 478-498.
Guala, M., Hommema, S.E., and Adrian, R.J. 2006,“Large-scale and very-large-scale motions in turbulent pipe
flow,” J Fluid Mech, Vol.554, pp. 521-542.
Hellström, L.H.O., and Smits, A.J., 2017, “Structure
identification in pipe flow using proper orthogonal
decomposition,”, Phil Trans R Soc A, Vol. 375, 20160086.
Lee, J.H., Sung, H.J., and Adrian, R.J. 2019, “Space-timeformation of very-large-scale motions in turbulent pipe flow,” J
Fluid Mech, Vol.881, pp. 1010-1047.
Monty, J.P., Stewart, J.A., Williams, R.C., and Chong,M.S., 2007, “Large-scale features in turbulent pipe and channel
flows,” J Fluids Mech, Vol. 589, pp. 147-156.
Piller, M., 2005, “Direct numerical simulation of turbulent
forced convection in a pipe,” Int J Numerical Methods in
Fluids, Vol. 49, pp. 583-602.

CONCLUSIONS
We have carried out well-resolved LESs for a turbulencepipe flow with circumferentially-varying heat flux, especiallypaying attention to the dissimilarity between temperature andvelocity fields; the previous DNSs (Antoranz et al., 2015)
suggested that corrections to the model with a constantturbulent Prandtl numbers are needed, implying thedissimilarity. The Reynolds numbers normalized with the pipe
radius and the friction velocities were set to 180 and 360,
corresponding to the previous DNSs. After confirming theagreement of turbulence statistic with previous DNSs and also
dependencies numerical parameters, such as grid resolutions,
SGS models, we investigated changes in turbulence
characteristics of temperature fields in the boundary layer due
to surface heat flux distributions and discussed differencesfrom velocity fields. The surface heat flux distributions, which
increase temperature fluctuations in the boundary layer,
activated intermittent entrainment motions and weakened
negative correlations between streamwise velocity withincreasing in characteristic length of temperature fields. Suchmodulation in large-scale structures of temperature fields must
be an origin of dissimilarities with velocity fields.

REFERENCES
Abe, K., 2013, “An improved anisotoropy-resolving
subgrid-scale model with the aid of a scale-similarity modelling
concept,” Int J Heat Fluid Flow, Vol. 29, pp. 42-52.
Ahn, J., and Sung, H.J., 2017, “Relationship between
streamwise and azimuthal length scales in a turbulent pipe
flow,” Phys Fluids, Vol. 29, 105112.
Antoranz, A., Gonzalo, A., Flores, O., and Garcia-Villalba,
M., 2015, “Numerical simulation of heat transfer in a pipe with
non-homogeneous thermal boundary conditions,” Int J Heat
Fluid Flow, Vol. 55, pp. 45-51.

5