In the present investigation, turbulent heat transfer in fully developed curved-pipe flow has been studied by using large eddy simulation (LES). We consider a fully developed turbulent curved-pipe flow with axially uniform wall heat flux. The friction Reynolds number under consideration is $Reτ$  = 1000 based on the mean friction velocity and the pipe radius, and the Prandtl number (Pr) is 0.71. To investigate the effects of wall curvature on turbulent flow and heat transfer, we varied the nondimensionalized curvature (δ) from 0.01 to 0.1. Dynamic subgrid-scale models for turbulent subgrid-scale stresses and heat fluxes were employed to close the governing equations. To elucidate the secondary flow structures due to the pipe curvature and their effect on the heat transfer, the mean quantities and various turbulence statistics of the flow and temperature fields are presented, and compared with those of the straight-pipe flow. The friction factor and the mean Nusselt number computed in the present study are in good agreement with the experimental results currently available in the literature. We also present turbulence intensities, skewness and flatness factors of temperature fluctuations, and cross-correlations of velocity and temperature fluctuations. In addition, we report the results of an octant analysis to clarify the correlation between near-wall turbulence structures and temperature fluctuation in the vicinity of the pipe wall. Based on our results, we attempt to clarify the effects of the pipe curvature on turbulent heat transfer. Our LES provides researchers and engineers with useful data to understand the heat-transfer mechanisms in turbulent curved-pipe flow, which has numerous applications in engineering.

## References

References
1.
Hawes
,
W. B.
,
1932
, “
Some Sidelights on the Heat Transfer Problem
,”
Trans. Inst. Chem. Eng.
,
10
, pp.
161
167
.
2.
Seban
,
R. A.
, and
McLaughlin
,
E. F.
,
1963
, “
Heat Transfer in Tube Coils With Laminar and Turbulent Flow
,”
Int. J. Heat Mass Transfer
,
6
(
5
), pp.
387
395
.
3.
Mori
,
Y.
, and
Nakayama
,
W.
,
1965
, “
Study on Forced Convective Heat Transfer in Curved Pipes (1st Report, Laminar Region)
,”
Int. J. Heat Mass Transfer
,
8
(
1
), pp.
67
82
.
4.
Mori
,
Y.
, and
Nakayama
,
W.
,
1967
, “
Study on Forced Convective Heat Transfer in Curved Pipes (2nd Report, Turbulent Region)
,”
Int. J. Heat Mass Transfer
,
10
(
1
), pp.
37
59
.
5.
Mori
,
Y.
, and
Nakayama
,
W.
,
1967
, “
Study on Forced Convective Heat Transfer in Curved Pipes (3rd Report, Theoretical Analysis Under the Condition of Uniform Wall Temperature and Practical Formulae)
,”
Int. J. Heat Mass Transfer
,
10
(
5
), pp.
681
695
.
6.
Pratt
,
N. H.
,
1947
, “
The Heat Transfer in a Reaction Tank Cooled by Means of a Coil
,”
Trans. Inst. Chem. Eng.
,
25
, pp.
163
180
.
7.
Rogers
,
G. F. C.
, and
Mayhew
,
Y. R.
,
1964
, “
Heat Transfer and Pressure Loss in Helically Coiled Tubes With Turbulent Flow
,”
Int. J. Heat Mass Transfer
,
7
(
11
), pp.
1207
1216
.
8.
Dravid
,
A. N.
,
Smith
,
K. A.
,
Merrill
,
E. W.
, and
Brian
,
P. L. T.
,
1971
, “
Effect of Secondary Fluid Motion on Laminar Flow Heat Transfer in Helically Coiled Tubes
,”
AIChE J.
,
17
(
5
), pp.
1114
1122
.
9.
Akiyama
,
M.
, and
Cheng
,
K. C.
,
1971
, “
Boundary Vorticity Method for Laminar Forced Convection Heat Transfer in Curved Pipes
,”
Int. J. Heat Mass Transfer
,
14
(
10
), pp.
1659
1675
.
10.
Akiyama
,
M.
, and
Cheng
,
K. C.
,
1972
, “
Laminar Forced Convection Heat Transfer in Curved Pipes With Uniform Wall Temperature
,”
Int. J. Heat Mass Transfer
,
15
(
7
), pp.
1426
1431
.
11.
Kalb
,
C. E.
, and
,
J. D.
,
1972
, “
Heat and Mass Transfer Phenomena for Viscous Flow in Curved Circular Tubes
,”
Int. J. Heat Mass Transfer
,
15
(
4
), pp.
801
817
.
12.
Kalb
,
C. E.
, and
,
J. D.
,
1974
, “
Fully Developed Viscous-Flow Heat Transfer in Curved Circular Tubes With Uniform Wall Temperature
,”
AIChE J.
,
20
(
2
), pp.
340
346
.
13.
Patankar
,
S. V.
,
Pratap
,
V. S.
, and
Spalding
,
D. B.
,
1974
, “
Prediction of Laminar Flow and Heat Transfer in Helically Coiled Pipes
,”
J. Fluid Mech.
,
62
(
3
), pp.
539
551
.
14.
Patankar
,
S. V.
,
Pratap
,
V. S.
, and
Spalding
,
D. B.
,
1975
, “
Prediction of Turbulent Flow in Curved Pipes
,”
J. Fluid Mech.
,
67
(
3
), pp.
583
595
.
15.
Zapryanov
,
Z.
,
Christov
,
C.
, and
Toshev
,
E.
,
1980
, “
Fully Developed Laminar Flow and Heat Transfer in Curved Tubes
,”
Int. J. Heat Mass Transfer
,
23
(
6
), pp.
873
880
.
16.
Xin
,
R. C.
, and
,
M. A.
,
1997
, “
The Effects of Prandtl Numbers on Local and Average Convective Heat Transfer Characteristics in Helical Pipes
,”
ASME J. Heat Transfer
,
119
(
3
), pp.
467
473
.
17.
Yang
,
G.
, and
,
M. A.
,
1996
, “
Turbulent Forced Convection in a Helicoidal Pipe With Substantial Pitch
,”
Int. J. Heat Mass Transfer
,
39
(
10
), pp.
2015
2022
.
18.
Lin
,
C. X.
, and
,
M. A.
,
1997
, “
Developing Turbulent Convective Heat Transfer in Helical Pipes
,”
Int. J. Heat Mass Transfer
,
40
(
16
), pp.
3861
3873
.
19.
Lin
,
C. X.
, and
,
M. A.
,
1999
, “
The Effects of Inlet Turbulence on the Development of Fluid Flow and Heat Transfer in a Helically Coiled Pipe
,”
Int. J. Heat Mass Transfer
,
42
(
4
), pp.
739
751
.
20.
Di Piazza
,
I.
, and
Ciofalo
,
M.
,
2010
, “
Numerical Prediction of Turbulent Flow and Heat Transfer in Helically Coiled Pipes
,”
Int. J. Therm. Sci.
,
49
(
4
), pp.
653
663
.
21.
Petukhov
,
B. S.
,
1970
, “
Heat Transfer and Friction in Turbulent Pipe Flow With Variable Physical Properties
,”
,
6
, pp.
503
564
.
22.
Di Liberto
,
M.
, and
Ciofalo
,
M.
,
2013
, “
A Study of Turbulent Heat Transfer in Curved Pipes by Numerical Simulation
,”
Int. J. Heat Mass Transfer
,
59
, pp.
112
125
.
23.
Boersma
,
B. J.
,
1997
, “
Electromagnetic Effects in Cylindrical Pipe Flow
,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
24.
Boersma
,
B. J.
, and
,
F. T. M.
,
1996
, “
Large-Eddy Simulation of Turbulent Flow in a Curved Pipe
,”
ASME J. Fluids Eng.
,
118
(
2
), pp.
248
254
.
25.
Germano
,
M.
,
Piomelli
,
U.
,
Moin
,
P.
, and
Cabot
,
W. H.
,
1991
, “
A Dynamic Subgrid-Scale Eddy Viscosity Model
,”
Phys. Fluids A
,
3
(
7
), pp.
1760
1765
.
26.
Cabot
,
W. H.
, and
Moin
,
P.
,
1993
, “
Large Eddy Simulation of Scalar Transport With the Dynamic Subgrid-Scale Model
,”
Large Eddy Simulation of Complex Engineering and Geophysical Flows
,
Cambridge University
, Cambridge, UK, Chap. 7.
27.
Akselvoll
,
K.
, and
Moin
,
P.
,
1995
, “
Large Eddy Simulation of Turbulent Confined Coannular Jets and Turbulent Flow Over a Backward Facing Step
,” Department of Mechanical Engineering, Stanford University, Technical Report No. TF-63.
28.
Lilly
,
D. K.
,
1992
, “
A Proposed Modification of the Germano Subgrid-Scale Closure Method
,”
Phys. Fluids A
,
4
(
3
), pp.
633
635
.
29.
Hüttl
,
T. J.
, and
Friedrich
,
R.
,
2000
, “
Influence of Curvature and Torsion on Turbulent Flow in Helically Coiled Pipes
,”
Int. J. Heat Fluid Flow
,
21
(
3
), pp.
345
353
.
30.
Hüttl
,
T. J.
, and
Friedrich
,
R.
,
2001
, “
Direct Numerical Simulation of Turbulent Flows in Curved and Helically Coiled Pipes
,”
Comput. Fluids
,
30
(
5
), pp.
591
605
.
31.
Akselvoll
,
K.
, and
Moin
,
P.
,
1996
, “
An Efficient Method for Temporal Integration of the Navier–Stokes Equations in Confined Axisymmetric Geometries
,”
J. Comput. Phys.
,
125
(
2
), pp.
454
463
.
32.
Kim
,
J.
, and
Moin
,
P.
,
1985
, “
Application of a Fractional-Step Method to Incompressible Navier–Stokes Equations
,”
J. Comput. Phys.
,
59
(
2
), pp.
308
323
.
33.
Patankar
,
S. V.
,
Liu
,
C. H.
, and
Sparrow
,
E. M.
,
1977
, “
Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area
,”
ASME J. Heat Transfer
,
99
(
2
), pp.
180
186
.
34.
Webster
,
D. R.
, and
Humphrey
,
J. A. C.
,
1993
, “
Experimental Observations of Flow Instability in a Helical Coil
,”
ASME J. Fluids Eng.
,
115
(
3
), pp.
436
443
.
35.
Webster
,
D. R.
, and
Humphrey
,
J. A. C.
,
1997
, “
Traveling Wave Instability in Helical Coil Flow
,”
Phys. Fluids
,
9
(
2
), pp.
407
418
.
36.
Cioncolini
,
A.
, and
Santini
,
L.
,
2006
, “
An Experimental Investigation Regarding the Laminar to Turbulent Flow Transition in Helically Coiled Pipes
,”
Exp. Therm. Fluid Sci.
,
30
(
4
), pp.
367
380
.
37.
Ito
,
H.
,
1959
, “
Friction Factors for Turbulent Flow in Curved Pipes
,”
J. Basic Eng.
,
81
(2), pp.
123
134
.
38.
,
L.
,
Ould-Rouiss
,
M.
, and
Lauriat
,
G.
,
2007
, “
Direct Numerical Simulation of Turbulent Heat Transfer in Pipe Flows: Effect of Prandtl Number
,”
Int. J. Heat Fluid Flow
,
28
(
5
), pp.
847
861
.
39.
Suzuki
,
H.
,
Suzuki
,
K.
, and
Sato
,
T.
,
1988
, “
Dissimilarity Between Heat and Momentum Transfer in a Turbulent Boundary Layer Disturbed by a Cylinder
,”
Int. J. Heat Mass Transfer
,
31
(
2
), pp.
259
265
.
40.
Volino
,
R. J.
, and
Simon
,
T. W.
,
1994
, “
An Application of Octant Analysis to Turbulent and Transitional Flow Data
,”
ASME J. Turbomach.
,
116
(
4
), pp.
752
758
.
41.
Eibeck
,
P. A.
, and
Eaton
,
J. K.
,
1987
, “
Heat Transfer Effects of a Longitudinal Vortex Embedded in a Turbulent Boundary Layer
,”
ASME J. Heat Transfer
,
109
(
1
), pp.
16
24
.
42.
Wroblewski
,
D. E.
, and
Eibeck
,
P. A.
,
1991
, “
Measurements of Turbulent Heat Transport in a Boundary Layer With an Embedded Streamwise Vortex
,”
Int. J. Heat Mass Transfer
,
34
(
7
), pp.
1617
1631
.
43.
Inaoka
,
K.
, and
Suzuki
,
K.
,
1995
, “
Structure of the Turbulent Boundary Layer and Heat Transfer Downstream of a Vortex Generator Attached to a LEBU Plate
,”
Turbulent Shear Flows
, Vol.
9
,
Springer
,
Berlin
, pp.
365
382
.
44.
Kasagi
,
N.
, and
Ohtsubo
,
Y.
,
1993
, “
Direct Numerical Simulation of Low Prandtl Number Thermal Field in a Turbulent Channel Flow
,”
Turbulent Shear Flows
, Vol.
8
,
Springer
,
Berlin, Germany
, pp.
97
119
.
45.
Choi
,
H. S.
,
Park
,
T. S.
, and
Suzuki
,
K.
,
2004
, “
Turbulence Characteristics of the Flows in a Wavy Channel
,”
Int. J. Transp. Phenom.
,
6
(
3
), pp.
197
212
.
46.
Choi
,
H. S.
, and
Suzuki
,
K.
,
2005
, “
Large Eddy Simulation of Turbulent Flow and Heat Transfer in a Channel With One Wavy Wall
,”
Int. J. Heat Fluid Flow
,
26
(
5
), pp.
681
694
.