The local (pointwise) entropy generation rate per unit volume S is a key to improving many energy processes and applications. Consequently, in the present study, the objectives are to examine the effects of Reynolds number and favorable streamwise pressure gradients on entropy generation rates across turbulent boundary layers on flat plates and—secondarily—to assess a popular approximate technique for their evaluation. About two-thirds or more of the entropy generation occurs in the viscous part, known as the viscous layer. Fundamental new results for entropy generation in turbulent boundary layers are provided by extending available direct numerical simulations. It was found that, with negligible pressure gradients, results presented in wall coordinates are predicted to be near “universal” in the viscous layer. This apparent universality disappears when a significant pressure gradient is applied; increasing the pressure gradient decreases the entropy generation rate. Within the viscous layer, the approximate evaluation of S differs significantly from the “proper” value but its integral, the entropy generation rate per unit surface area Sap, agrees within 5% at its edge.

1.
Bejan
,
A.
, 1982,
Entropy Generation Through Heat and Fluid Flow
,
Wiley
,
New York
.
2.
Onsager
,
L.
, 1931, “
Reciprocal Relations in Irreversible Processes. I
,”
Phys. Rev.
0031-899X,
37
, pp.
405
426
.
3.
Prigogine
,
I.
, 1978, “
Time, Structure and Fluctuations
,”
Science
0036-8075,
201
, pp.
777
785
.
4.
Kock
,
F.
, and
Herwig
,
H.
, 2005, “
Entropy Production Calculation for Turbulent Shear Flows and Their Implementation in CFD Codes
,”
Int. J. Heat Fluid Flow
0142-727X,
26
, pp.
672
680
.
5.
Kestin
,
J.
, 1980, “Availability: The Concept and Associated Terminology,” Energy Intl. J., 5, pp. 679–692.
6.
Gouy
,
M.
, 1889, “Sur l’energie Utilisable,” J. Phys., 8, pp. 501–518.
7.
Stodola
,
A.
, 1910,
Steam and Gas Turbines
,
McGraw-Hill
,
New York
.
8.
Clausius
,
R.
, 1887,
Die Mechanische Wärmetheorie
,
Vieweg
,
Braunschweig
.
9.
Lichty
,
L. C.
, 1936,
Thermodynamics
,
McGraw-Hill
,
New York
.
10.
Bradshaw
,
P.
, 1975,
An Introduction to Turbulence and Its Measurement
, 2nd ed.,
Pergamon
,
Oxford
.
11.
Rotta
,
J. C.
, 1962, “
Turbulent Boundary Layers in Incompressible Flow
,”
Progress in Aeronautical Sciences
,
A.
Ferri
,
D.
Küchemann
, and
L. H. G.
Sterne
, eds.,
Pergamon
,
Oxford
, Vol.
2
, pp.
1
219
.
12.
Spalart
,
P. R.
, 1986, “
Numerical Study of Sink-Flow Boundary Layers
,”
J. Fluid Mech.
0022-1120,
172
, pp.
307
328
.
13.
Spalart
,
P. R.
, 1988, “
Direct Simulation of a Turbulent Boundary Layer up to Reθ=1410
,”
J. Fluid Mech.
0022-1120,
187
, pp.
61
98
.
14.
London
,
A. L.
, 1960,
Advanced Thermodynamics
,
Stanford University
,
Palo Alto, CA.
, Course ME 233.
15.
Reynolds
,
W. C.
, and
Perkins
,
H. C.
, 1970,
Engineering Thermodynamics
,
McGraw-Hill
,
New York
.
16.
Cebeci
,
T.
, and
Bradshaw
,
P.
, 1984,
Physical and Computational Aspects of Convective Heat Transfer
,
Springer
,
New York
.
17.
Gersten
,
K.
, and
Herwig
,
H.
, 1992,
Strömungsmechanik
,
Vieweg
,
Braunschweig
.
18.
Schlichting
,
H.
, and
Gersten
,
K.
, 1997,
Grenzschicht-Theorie, 9., Völlig Neubearbeitete und Erweiterte Auflage
,
Springer
,
Berlin
.
19.
Schlichting
,
H.
, and
Gersten
,
K.
, 2000,
Boundary Layer Theory
, 8th revised ed.,
Springer
,
Berlin
.
20.
Brodkey
,
R. S.
,
Nychas
,
S. G.
,
Taraba
,
J. L.
, and
Wallace
,
J. M.
, 1973, “
Turbulent Energy Production, Dissipation and Transfer
,”
Phys. Fluids
0031-9171,
16
, pp.
2010
2011
.
21.
Brodkey
,
R. S.
,
Taraba
,
J. L.
,
Nychas
,
S. G.
, and
Wallace
,
J. M.
, 1974, “
Reply To Comments By P. Bradshaw
,”
Phys. Fluids
0031-9171,
17
, p.
2150
.
22.
Bradshaw
,
P.
, 1974, “
Comments on Turbulent Energy Production, Dissipation and Transfer
,”
Phys. Fluids
0031-9171,
17
, pp.
2149
.
23.
Kock
,
F.
, and
Herwig
,
H.
, 2004, “
Local Entropy Production in Turbulent Shear Flows: A High-Reynolds Number Model With Wall Functions
,”
Int. J. Heat Mass Transfer
0017-9310,
47
, pp.
2205
2215
.
24.
Schlichting
,
H.
, 1968,
Boundary Layer Theory
, 6th ed.
McGraw-Hill
,
New York
.
25.
Kline
,
S. J.
,
Reynolds
,
W. C.
,
Schraub
,
F. A.
, and
Rundstadler
,
P. W.
, 1967, “
The Structure of Turbulent Boundary Layers
,”
J. Fluid Mech.
0022-1120,
30
, pp.
741
773
.
26.
McEligot
,
D. M.
, and
Eckelmann
,
H.
, 2006, “
Laterally Converging Duct Flows. Part 3. Mean Turbulence Structure in the Viscous Layer
,”
J. Fluid Mech.
0022-1120,
549
, pp.
25
59
.
27.
Julien
,
H. L.
,
Kays
,
W. M.
, and
Moffat
,
R. J.
, 1969, “
The Turbulent Boundary Layer on a Porous Plate: Experimental Study of the Effects of A Favorable Pressure Gradient
,”
Stanford University
, Palo Alto, Technical Report HMT-4.
28.
Finnicum
,
D. S.
, and
Hanratty
,
T. J.
, 1988, “
Effect of Favorable Pressure Gradients on Turbulent Boundary Layers
,”
AIChE J.
0001-1541,
34
, pp.
529
540
.
29.
Nieuwstadt
,
F. T. M.
, and
Bradshaw
,
P.
, 1997, “
Similarities and Differences in Turbulent Boundary-Layer, Pipe and Channel Flows
,”
Boundary-Layer Separation in Aircraft Aerodynamics
,
R. A. W. M.
Henkes
and
P. G.
Bakker
, eds.,
Delft University Press
,
Delft
, pp.
15
22
.
30.
Nickels
,
T. B.
, 2004, “
Inner Scaling for Wall-Bounded Flows Subject to Large Pressure Gradients
,”
J. Fluid Mech.
0022-1120,
521
, pp.
217
239
.
31.
Klebanoff
,
P. S.
, 1955, “
Characteristics of Turbulence in a Boundary Layer With Zero Pressure Gradient
,” NACA Report 1247, pp.
1135
1153
.
32.
deGraaff
,
D. B.
, and
Eaton
,
J. K.
, 2000, “
Reynolds-Number Scaling of the Flat-Plate Turbulent Boundary Layer
,”
J. Fluid Mech.
0022-1120,
422
, pp.
319
346
.
33.
Laadhari
,
F.
, 2007, “
Reynolds Number Effect on the Dissipation Function in Wall-Bounded Flows
,”
Phys. Fluids
1070-6631,
19
(
3
), pp.
038101
-1–038101-
4
.
34.
Jovanovic
,
J.
,
Pashtrapanska
,
M.
,
Frohnapfel
,
B.
,
Durst
,
F.
,
Koskinen
,
J.
, and
Koskinen
,
K.
, 2006, “
On the Mechanism Responsible for Turbulent Drag Reduction by Dilute Addition of High Polymers: Theory, Experiments, Simulations and Predictions
,”
ASME J. Fluids Eng.
0098-2202,
128
, pp.
118
130
.
35.
Antonia
,
R. A.
,
Kim
,
J.
, and
Browne
,
L. W. B.
, 1991, “
Some Characteristics of Small-Scale Turbulence in a Turbulent Duct Flow
,”
J. Fluid Mech.
0022-1120,
233
, pp.
369
388
.
36.
McEligot
,
D. M.
,
Nolan
,
K. P.
,
Walsh
,
E. J.
, and
Laurien
,
E.
, 2008, “
Effect of Pressure Gradients on Entropy Generation in the Viscous Layers of Turbulent Wall Flows
,”
Int. J. Heat Mass Transfer
0017-9310,
51
, pp.
1104
1114
.
37.
Abe
,
H.
,
Kawamura
,
H.
, and
Matsuo
,
Y.
, 2001, “
Direct Numerical Simulation of a Fully Developed Turbulent Channel Flow With Respect to the Reynolds Number Dependence
,”
ASME J. Fluids Eng.
0098-2202,
123
, pp.
382
393
.
38.
Tsukahara
,
T.
,
Seki
,
Y.
,
Kawamura
,
H.
, and
Tochio
,
D.
, 2005, “
DNS of Turbulent Channel Flow at Very Low Reynolds Numbers
,”
Proceedings of the Fourth International Symposium Turbulent Shear Flow Phenomena
,
Williamsburg, VA
, Jun. 27–29, pp.
935
940
.
39.
Skare
,
P. E.
, and
Krogstad
,
P.-A.
, 1994, “
A Turbulent Equilibrium Boundary Layer Near Separation
,”
J. Fluid Mech.
0022-1120,
272
, pp.
319
348
.
40.
Aubertine
,
C. D.
, and
Eaton
,
J. K.
, 2005, “
Turbulence Development in a Non-Equilibrium Turbulent Boundary Layer With Mild Adverse Pressure Gradient
,”
J. Fluid Mech.
0022-1120,
532
, pp.
345
364
.
41.
Jones
,
M. B.
,
Marusic
,
I.
, and
Perry
,
A. E.
, 2001, “
Evolution and Structure of Sink-Flow Turbulent Boundary Layers
,”
J. Fluid Mech.
0022-1120,
428
, pp.
1
27
.
42.
Jones
,
W. P.
, and
Launder
,
B. E.
, 1972, “
Some Properties of Sink Flow Turbulent Boundary Layers
,”
J. Fluid Mech.
0022-1120,
56
, pp.
337
351
.
43.
Hinze
,
J. O.
, 1975,
Turbulence
, 2nd ed.,
McGraw-Hill
,
New York
.
44.
Wilcox
,
D. C.
, 1998,
Turbulence Modeling for CFD
, 2nd ed.,
DCW Industries
,
La Canada, CA
.
45.
Kim
,
J.
,
Moin
,
P.
, and
Moser
,
R. D.
, 1987, “
Turbulent Statistics in Fully Developed Channel Flow at Low Reynolds Number
,”
J. Fluid Mech.
0022-1120,
177
, pp.
133
166
.
46.
Bradshaw
,
P.
, and
Perot
,
J. B.
, 1993, “
A Note on Turbulent Energy Dissipation in the Viscous Wall Region
,”
Phys. Fluids A
0899-8213,
5
(
12
), pp.
3305
3306
.
47.
Moser
,
R. D.
, and
Moin
,
P.
, 1987, “
Direct Numerical Simulation of Curved Turbulent Channel Flow
,”
J. Fluid Mech.
0022-1120,
175
, pp.
479
510
, also NASA TM-85974.
48.
Narasimha
,
R.
, and
Sreenivasan
,
K. R.
, 1979, “
Relaminarization of Fluid Flows
,”
Adv. Appl. Mech.
0065-2156,
19
, pp.
221
309
.
49.
McEligot
,
D. M.
, and
Eckelmann
,
H.
, 2003, “
Effects of Laterally Converging Flows on Mean Turbulence Structure in the Viscous Layer
,”
Idaho National Engineering and Environmental Laboratory
, Technical Report INEEL/EXT-2002-697.
50.
Senecal
,
V. E.
, 1952, “
Fluid Flow in the Transition Zone
,” Carnegie Institute of Technology, Ph.D. thesis.
51.
Patel
,
V. C.
, 1965, “
Calibration of the Preston Tube and Limitations on Its Use in Pressure Gradients
,”
J. Fluid Mech.
0022-1120,
23
, pp.
185
208
.
52.
McEligot
,
D. M.
,
Ormand
,
L. W.
, and
Perkins
,
H. C.
, 1966, “
Internal Low Reynolds Number Turbulent and Transitional Gas Flow With Heat Transfer
,”
ASME J. Heat Transfer
0022-1481,
88
, pp.
239
245
.
53.
Jovanovic
,
J.
, and
Hillerbrand
,
R.
, 2005, “
On Peculiar Property of the Velocity Fluctuations in Wall-Bounded Flows
,”
J. Therm. Sci.
1003-2169,
9
(
1
), pp.
3
12
.
54.
Moore
,
J.
, and
Moore
,
J. G.
, 1983, “
Entropy Production Rates From Viscous Flow Calculations. Part I. A Turbulent Boundary Layer Flow
,” ASME Paper No. 83-GT-70.
55.
O’Donnell
,
F. K.
, and
Davies
,
M. R. D.
, 1999, “
Measurements of Turbine Blade Aerodynamic Entropy Generation Rate
,”
Third European Conference Turbomachinery
,
London
, pp.
43
53
.
56.
Stieger
,
R. D.
, and
Hodson
,
H. P.
, 2003, “
Unsteady Dissipation Measurements on a Flat Plate Subject to Wake Passing
,”
Proc., Inst. Mech. Eng., Part A
0957-6509,
217
, pp.
413
419
.
57.
Hyhlik
,
T.
, and
Marsik
,
F.
, 2006, “
New Approach to Turbulence Model Testing Based on the Entropy Production and the Analysis of Simple Wall Flows
,”
Turbulence, Heat And Mass Transfer 5
,
K.
Hanjalic
,
Y.
Nagano
, and
S.
Jakirlic
, eds.,
Begell House
,
New York
, pp.
377
380
.
58.
Tanaka
,
T.
, and
Eaton
,
J. K.
, 2007, “
A Correction Method for Measuring Turbulence Kinetic Energy Dissipation Rate by PIV
,”
Exp. Fluids
0723-4864,
42
, pp.
893
902
.
59.
Huffman
,
G. D.
, and
Bradshaw
,
P.
, 1972, “
A Note on Von Karman’s Constant in Low Reynolds Number Turbulent Flows
,”
J. Fluid Mech.
0022-1120,
53
, pp.
45
60
.
60.
McEligot
,
D. M.
, 1985, “
Measurement of Wall Shear Stress in Favorable Pressure Gradients
,”
Lect. Notes Phys.
0075-8450,
235
, pp.
292
303
.
61.
George
,
W. K.
, and
Hussein
,
H. J.
, 1991, “
Locally Axisymmetric Turbulence
,”
J. Fluid Mech.
0022-1120,
233
, pp.
369
388
.
62.
Khoo
,
B. C.
,
Chew
,
Y. T.
, and
Teo
,
C. J.
, 2000, “
On Near-Wall Hot-Wire Measurements
,”
Exp. Fluids
0723-4864,
29
, pp.
448
460
.
63.
Jovanovic
,
J.
, 2006, Personal electronic communication, Lehrstuhl für Strömungsmechanik, Uni. Erlangen, Oct. 11.
64.
Frohnapfel
,
B.
,
Lammers
,
P.
,
Jovanovic
,
J.
, and
Durst
,
F.
, 2007, “
Interpretation of the Mechanism Associated With Turbulent Drag Reduction in Terms of Anisotropy Invariants
,”
J. Fluid Mech.
0022-1120,
577
, pp.
457
466
.
65.
Dimitropoulos
,
C. D.
,
Sureshkumar
,
R.
, and
Beris
,
A. N.
, 1998, “
Direct Numerical Simulation of Viscoelastic Turbulent Channel Flow Exhibiting Drag Reduction: Effects of the Variation of Rheological Parameters
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
79
, pp.
433
468
.
You do not currently have access to this content.