Abstract

Fourier’s law, which forms the basis of most engineering prediction methods for the turbulent heat fluxes, is known to fail badly in capturing the effects of streamline curvature on the rate of heat transfer in turbulent shear flows. In this paper, an alternative model, which is both algebraic and explicit in the turbulent heat fluxes and which has been formulated from tensor-representation theory, is presented, and its applicability is extended by incorporating the effects of a wall on the turbulent heat transfer processes in its vicinity. The model’s equations for flows with curvature in the plane of the mean shear are derived and calculations are performed for a heated turbulent boundary layer, which develops over a flat plate before encountering a short region of high convex curvature. The results show that the new model accurately predicts the significant reduction in the wall heat transfer rates wrought by the stabilizing-curvature effects, in sharp contrast to the conventional model predictions, which are shown to seriously underestimate the same effects. Comparisons are also made with results from a complete heat-flux transport model, which involves the solution of differential transport equations for each component of the heat-flux tensor. Downstream of the bend, where the perturbed boundary layer recovers on a flat wall, the comparisons show that the algebraic model yields indistinguishable predictions from those obtained with the differential model in regions where the mean-strain field is in rapid evolution and the turbulence processes are far removed from local equilibrium.

1.
Patankar
,
S. V.
,
Pratap
,
V. S.
, and
Spalding
,
D. B.
, 1975, “
Prediction of Turbulent Flow in Curved Pipes
,”
J. Fluid Mech.
0022-1120,
67
, pp.
583
595
.
2.
Howard
,
J. H. J.
,
Patankar
,
S. V.
, and
Bordynuik
,
R. M.
, 1980, “
Flow Prediction in Rotating Ducts Using Coriolis-Modified Turbulence Models
,”
ASME J. Fluids Eng.
0098-2202,
102
(
4
), pp.
456
461
.
3.
Berhe
,
M. K.
, and
Patankar
,
S. V.
, 1999, “
Curvature Effects on Discrete-Hole Film Cooling
,”
ASME J. Turbomach.
0889-504X,
141
(
4
), pp.
781
791
.
4.
Berhe
,
M. K.
, and
Patankar
,
S. V.
, 1999, “
Investigation of Discrete-Hole Film Cooling Parameters Using Curved-Plate Models
,”
ASME J. Turbomach.
0889-504X,
141
(
4
), pp.
792
803
.
5.
Rayleigh
,
T. W. S.
, 1917, “
Dynamics of Revolving Fluids
,”
Proc. R. Soc. London, Ser. A
0950-1207,
93
, pp.
148
154
.
6.
Bradshaw
,
P.
, 1973, “
Effects of Streamwise Curvature on Turbulent Flows
,” AGARDograph No. 169.
7.
Simon
,
T. W.
, and
Moffat
,
R. J.
, 1979, “
Heat Transfer Through Turbulent Boundary Layers—The Effects of Introduction of and Recovery From Convex Curvature
,” ASME Winter Annual Meeting, New York, ASME Paper No. 79-WA/GT-10.
8.
Gibson
,
M. M.
,
Verriopoulos
,
C. A.
, and
Nagano
,
Y.
, 1982, “
Measurements in the Heated Turbulent Boundary Layer on a Mildly Curved Convex Surface
,”
Turbulent Shear Flows
,
Springer-Verlag
,
Berlin
, Vol.
3
.
9.
Gibson
,
M. M.
, 1978, “
An Algebraic Stress and Heat-Flux Model for Turbulent Shear Flow With Streamline Curvature
,”
Int. J. Heat Mass Transfer
0017-9310,
21
, pp.
1609
1617
.
10.
Younis
,
B. A.
, and
Berger
,
S. A.
, 2006, “
On Predicting the Effects of Streamline Curvature on the Turbulent Prandtl Number
,”
ASME J. Appl. Mech.
0021-8936,
73
, pp.
1
6
.
11.
Smits
,
A. J.
,
Young
,
S. T. P.
, and
Bradshaw
,
P.
, 1979, “
The Effect of Short Regions of High Surface Curvature on Turbulent Boundary Layers
,”
J. Fluid Mech.
0022-1120,
94
, pp.
209
242
.
12.
Younis
,
B. A.
,
Speziale
,
C. G.
, and
Clark
,
T. T.
, 2005, “
A Rational Model for the Turbulent Scalar Fluxes
,”
Proc. R. Soc. London, Ser. A
1364-5021,
461
, pp.
575
594
.
13.
Kaltenbach
,
H.-J.
,
Gerz
,
T.
, and
Schumann
,
U.
, 1994, “
Large-Eddy Simulation of Homogeneous Turbulence and Diffusion in Stably Stratified Shear Flow
,”
J. Fluid Mech.
0022-1120,
280
, pp.
1
40
.
14.
Weigand
,
B.
,
Schwartzkopff
,
T.
, and
Sommer
,
T. P.
, 2002, “
A Numerical Investigation of the Heat Transfer in a Parallel Plate Channel With Piecewise Constant Wall Temperature Boundary Condition
,”
ASME J. Heat Transfer
0022-1481,
124
, pp.
626
634
.
15.
Kim
,
J.
,
Moin
,
P.
, and
Moser
,
R.
, 1987, “
Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number
,”
J. Fluid Mech.
0022-1120,
177
, pp.
133
166
.
16.
Pope
,
S. B.
, 2000,
Turbulent Flows
,
Cambridge University Press
,
Cambridge, England
.
17.
Abe
,
H.
, and
Kawamura
,
H.
, 2002, “
A Study of Turbulence Thermal Structure in a Channel Flow Through DNS Up to Reτ=640 with Pr=0.71
,”
Proc. 9th European Turbulence Conference
,
University of Southampton
, Southampton, pp.
399
402
.
18.
Kawamura
,
H.
,
Ohsaka
,
K.
,
Abe
,
H.
, and
Yamamoto
,
K.
, 1998, “
DNS of Turbulent Heat Transfer in Channel Flow With Low to Medium-High Prandtl Number
,”
Int. J. Heat Fluid Flow
0142-727X,
19
, pp.
482
491
.
19.
Kawamura
,
H.
,
Abe
,
H.
, and
Shingai
,
K.
, 2002, “
DNS of Turbulence and Heat Transport in a Channel Flow With Different Reynolds and Prandtl Numbers and Boundary Conditions
,”
Proc. 3rd Int. Symp. on Turbulence, Heat and Mass Transfer
,
Aichi Shuppan
, Japan, pp.
15
32
.
20.
Debusschere
,
B.
, and
Rutland
,
C. J.
, 2003, “
Turbulent Scalar Transport Mechanisms in Plane Channel and Couette Flows
,”
Int. J. Heat Mass Transfer
0017-9310,
47
, pp.
222
242
.
21.
Kasagi
,
N.
, and
Iida
,
O.
, 1999, “
Progress in Direct Numerical Simulation of Turbulent Heat Transfer
,”
Proc 5th ASME/JSME Joint Thermal Engineering Conference
, San-Diego, ASME,
New York
, pp.
1
17
.
22.
Gibson
,
M. M.
,
Jones
,
W. P.
, and
Younis
,
B. A.
, 1981, “
Calculation of Turbulent Boundary Layers Over Curved Surfaces
,”
Phys. Fluids
0031-9171,
24
, pp.
386
395
.
23.
Gibson
,
M. M.
, and
Younis
,
B. A.
, 1982, “
Modeling the Curved Turbulent Wall Jet
,”
AIAA J.
0001-1452,
20
(
12
), pp.
1707
1712
.
24.
Gibson
,
M. M.
, and
Younis
,
B. A.
, 1986, “
Calculation of Swirling Jets With a Reynolds Stress Closure
,”
Phys. Fluids
0031-9171,
29
, pp.
38
48
.
25.
Gibson
,
M. M.
, and
Younis
,
B. A.
, 1986, “
Calculation of Boundary Layers With Sudden Transverse Strains
,”
ASME J. Fluids Eng.
0098-2202,
108
, pp.
470
475
.
26.
Malin
,
M. R.
, and
Younis
,
B. A.
, 1990, “
Calculation of Turbulent Buoyant Plumes With a Reynolds-Stress and Heat-Flux Transport Closure
,”
Int. J. Heat Mass Transfer
0017-9310,
33
(
10
), pp.
2247
2264
.
27.
Gibson
,
M. M.
, and
Launder
,
B. E.
, 1978, “
Ground Effects on Pressure Fluctuations in the Atmospheric Boundary Layer
,”
J. Fluid Mech.
0022-1120,
86
(
3
), pp.
491
511
.
28.
Younis
,
B. A.
, 1996, “
EXPRESS: Accelerated Parabolic Reynolds Stress Solver
,” Hydraulics Section Report, City University, London.
29.
Patankar
,
S. V.
, and
Spalding
,
D. B.
, 1970,
Heat and Mass Transfer in Boundary Layers
,
Intertext Books
,
London
.
30.
Gibson
,
M. M.
, and
Servat-Djoo
,
K.
, 1989, “
Effect of a Short Region of High Convex Curvature on Heat Transfer Through a Turbulent Boundary Layer
,”
Int. J. Heat Fluid Flow
0142-727X,
10
(
1
), pp.
75
82
.
31.
Rubinstein
,
R.
, and
Barton
,
J. M.
, 1991, “
Renormalization Group Analysis of Anisotropic Diffusion in Turbulent Shear Flows
,”
Phys. Fluids A
0899-8213,
3
, pp.
415
421
.
32.
Rogers
,
M. M.
,
Mansour
,
N. N.
, and
Reynolds
,
W. C.
, 1989, “
An Algebraic Model for the Turbulent Flux of a Passive Scalar
,”
J. Fluid Mech.
0022-1120,
203
, pp.
77
101
.
33.
Suga
,
K.
, and
Abe
,
K.
, 1999, “
Nonlinear Eddy Viscosity Modelling for Turbulence and Heat Transfer Near Wall and Shear-Free Boundaries
,”
Int. J. Heat Fluid Flow
0142-727X,
39
, pp.
455
465
.
34.
So
,
R. M. C.
, and
Sommer
,
T. P.
, 1996, “
An Explicit Algebraic Heat-Flux Model for the Temperature Field
,”
Int. J. Heat Mass Transfer
0017-9310,
39
, pp.
455
465
.
You do not currently have access to this content.