The mean velocity profile and friction factor in turbulent flows with polymer additives are investigated using Prandtl’s mixing-length theorem. This study reveals that the mixing-length theorem is valid to express the drag-reducing phenomenon and that the presence of polymer additives increases the damping factor B in van Driest’s model; subsequently reducing the mixing-length, this interprets that the polymer hampers the transfer of turbulent momentum flux, the velocity is increased, and flow drag is reduced. This study also discusses the onset Reynolds number for drag reduction to occur. The predicted velocity, friction factor, and onset Reynolds number are in good agreement with the measured data in the literature.

1.
Toms
,
B. A.
, 1949, “
Some Observation on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers
,”
Proceedings of the First International Congress on Rheology
, Vol.
2
, pp.
135
141
.
2.
Virk
,
P. S.
, 1971, “
An Elastic Sublayer Model for Drag Reduction by Dilute Solutions of Linear Macromolecules
,”
J. Fluid Mech.
0022-1120,
45
, pp.
417
440
.
3.
Larson
,
R. G.
, 2003, “
Analysis of Polymer Turbulent Drag Reduction in Flow Past a Flat Plate
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
111
(
2–3
), pp.
229
250
.
4.
Min
,
T.
,
Choi
,
H.
, and
Yoo
,
J. Y.
, 2003, “
Maximum Drag-Reduction in a Turbulent Channel Flow by Polymer Additives
,”
J. Fluid Mech.
0022-1120,
492
, pp.
91
100
.
5.
Min
,
T.
,
Yoo
,
J. Y.
,
Choi
,
H.
, and
Joseph
,
D. D.
, 2003, “
Drag Reduction by Polymer Additives in a Turbulent Channel Flow
,”
J. Fluid Mech.
0022-1120,
486
, pp.
213
238
.
6.
Gasljevic
,
K.
Aguilar
,
G.
, and
Matthys
,
E. F.
, 2001, “
On Two Distinct Types of Drag-Reducing Fluids, Diameter Scaling, and Turbulent Profiles
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
96
(
3
), pp.
405
425
.
7.
Ryskin
,
G.
, 1987, “
Turbulent Drag Reduction by Polymers: A Quantitative Theory
,”
Phys. Rev. Lett.
0031-9007,
59
, pp.
2059
2062
.
8.
Warholic
,
M. D.
,
Heist
,
D. K.
,
Katcher
,
M.
, and
Hanratty
,
T. J.
, 2001, “
A Study With Particle-Image Velocimetry of the Influence of Drag-Reducing Polymers on the Structure of Turbulence
,”
Exp. Fluids
0723-4864,
31
, pp.
474
483
.
9.
White
,
C. M.
,
Somandepalli
,
V. S. R.
, and
Mungal
,
M. G.
, 2004, “
The Turbulence Structure of Drag-Reduced Boundary Layer Flow
,”
Exp. Fluids
0723-4864,
36
, pp.
62
69
.
10.
Durst
,
F.
,
Keck
,
T.
, and
Kline
,
R.
, 1985, “
Turbulence Quantities and Reynolds Stresses in Pipe Flow of Polymer Solutions
,”
Proceedings of the First International Conference on Laser Anemometer-Advances and Applications
, BHRA, p.
31
.
11.
Willmarth
,
W. W.
,
Wei
,
T.
, and
Lee
,
C. O.
, 1987, “
Laser Anemometer Measurements of Reynolds Stress in a Turbulent Channel Flow With Drag Reducing Polymer Additives
,”
Phys. Fluids
0031-9171,
30
, pp.
933
.
12.
Luchik
,
T. S.
, and
Tierderman
,
W. G.
, 1988, “
Turbulent Structure in Low-Concentration Drag-Reducing Channel Flows
,”
J. Fluid Mech.
0022-1120,
190
, pp.
241
263
.
13.
Harder
,
K. J.
, and
Tiederman
,
W. G.
, 1991, “
Drag Reduction and Turbulent Structure in Two-Dimensional Channel Flows
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
336
, pp.
19
34
.
14.
Wei
,
T.
, and
Willmarth
,
W. W.
, 1992, “
Modifying Turbulent Structure With Drag-Reducing Polymer Additives in Turbulent Channel Flows
,”
J. Fluid Mech.
0022-1120,
245
, pp.
619
641
.
15.
Lumley
,
J. L.
, 1969, “
Drag Reduction by Additives
,”
Annu. Rev. Fluid Mech.
0066-4189,
1
, pp.
367
384
.
16.
L’vov
,
V. S.
,
Pomyalov
,
A.
,
Procaccia
,
I.
, and
Tiberkvich
,
V.
, 2004, “
Drag Reduction by Polymers in Wall Bounded Turbulence
,”
Phys. Rev. Lett.
0031-9007,
92
, p.
244503
-4.
17.
Den Toonder
,
J. M. J.
,
Hulsen
,
M. A.
,
Kuiken
,
G. D. C.
, and
Nieuwstadt
,
F. T. M.
, 1997, “
Drag Reduction by Polymer Additives in a Turbulent Pipe Flow: Numerical and Laboratory Experiments
,”
J. Fluid Mech.
0022-1120,
337
, pp.
193
231
.
18.
Warholic
,
M. D.
,
Massah
,
H.
, and
Hanratty
,
T. J.
, 1999, “
Influence of Drag-Reducing Polymers on Turbulence: Effects of Reynolds Number, Concentration and Mixing
,”
Exp. Fluids
0723-4864,
27
, pp.
461
472
.
19.
Gyr
,
A.
, and
Tsinober
,
A.
, 1997, “
On the Rheological Nature of Drag Reduction Phenomena
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
73
, pp.
153
162
.
20.
Schümmer
,
P.
, and
Thielen
,
W.
, 1980, “
Structure of Turbulence in Viscoelastic Fluids
,”
Chem. Eng. Commun.
0098-6445,
4
, pp.
593
606
.
21.
Giesekus
,
H.
, 1981,
Structure of Turbulence in Drag Reducing Fluids
,
Lecture Series 1981-86
,
von Karman Institute for Fluid Dynamics
,
Rhode-Sain-Genèse, Belgium
.
22.
Yang
,
S. Q.
, and
Dou
,
G.
, 2005, “
Drag Reduction in Flat-Plate Turbulent Boundary Layer Flow by Polymer Additive
,”
Phys. Fluids
1070-6631,
17
(
6
), p.
065104
.
23.
Dou
,
G. R.
, 1996, “
Basic Law in Mechanics of Turbulent Flows
,”
China Ocean Eng.
0890-5487,
10
(
1
), pp.
1
44
.
24.
Berman
,
N. S.
, 1989,
Drag Reduction in Fluids Flows: Techniques for Friction Control
,
R. H. J.
Sellin
and
R. T.
Moses
, eds.,
Ellis Horwood Limited
,
Chichester
.
25.
Ptasinski
,
P. K.
,
Boersma
,
B. J.
,
Nieuwstadt
,
F. T. M.
,
Hulsen
,
M. A.
,
Van den Brule
,
B. H. A. A.
, and
Hunt
,
J. C. R.
, 2003, “
Turbulent Channel Flow Near Maximum Drag Reduction: Simulations, Experiments and Mechanisms
,”
J. Fluid Mech.
0022-1120,
490
, pp.
251
291
.
26.
Benzi
,
R.
,
Angelis
,
E. D.
,
L’vov
,
V. S.
,
Procaccia
,
I.
, and
Tiberkevich
,
V.
, 2006, “
Maximum Drag Reduction Asymptotes and the Cross-Over to the Newtonian Plug
,”
J. Fluid Mech.
0022-1120,
551
, pp.
185
195
.
27.
Van Driest
,
E. R.
, 1956, “
On Turbulent Flow Near a Wall
,”
J. Aeronaut. Sci.
0095-9812,
23
, pp.
1007
1010
.
28.
Hinze
,
J. O.
, 1975,
Turbulence
,
2nd ed.
,
McGraw-Hill
,
New York
.
29.
Cebeci
,
T.
, and
Smith
,
A. M. O.
, 1974,
Analysis of Turbulent Boundary Layers
,
Academic
,
New York
, Chap. 6.
30.
White
,
C. M.
, and
Mungal
,
M. G.
, 2008, “
Mechanics and Prediction of Turbulent Drag Reduction With Polymer Additives
,”
Annu. Rev. Fluid Mech.
0066-4189,
40
, pp.
235
256
.
31.
Gyr
,
A.
, and
Bewersdorff
,
H. W.
, 1995,
Drag Reduction of Turbulent Flow by Additives
,
Kluwer Academic
,
Dordrecht
.
32.
Hinch
,
E. J.
, 1977, “
Mechanical Models of Dilute Polymer Solutions in Strong Flows
,”
Phys. Fluids
0031-9171,
20
, pp.
22
30
.
33.
Metzner
,
A. B.
, and
Metzner
,
A. P.
, 1970, “
Stress Level in Rapid Extensional Flow of Polymeric Fluids
,”
Rheol. Acta
0035-4511,
9
, pp.
174
181
.
34.
Lumley
,
J. L.
, 1973, “
Drag Reduction in Turbulent Flow by Polymer Additives
,”
J. Polym. Sci. Macromol. Rev.
0076-2083,
7
, pp.
263
290
.
35.
Ptasinski
,
P. K.
,
Nieuwstadt
,
F. T. M.
,
Van Den Brule
,
B. H. A. A.
, and
Hulsen
,
M. A.
, 2001, “
Experiments in Turbulent Pipe Flow With Polymer Additives at Maximum Drag Reduction
,”
Flow, Turbul. Combust.
1386-6184,
66
, pp.
159
182
.
36.
De Angelis
,
E.
,
Casciola
,
C. M.
, and
Piva
,
R.
, 2002, “
DNS of Wall Turbulence: Dilute Polymers and Self-Sustaining Mechanisms
,”
Comput. Fluids
0045-7930,
31
, pp.
495
507
.
37.
Housiadas
,
K. D.
, and
Beris
,
A. N.
, 2003, “
Polymer-Induced Drag Reduction: Effects of the Variations in Elasticity and Inertia in Turbulent Viscoelastic Channel Flow
,”
Phys. Fluids
1070-6631,
15
, pp.
2369
2384
.
38.
Reischman
,
M. M.
, and
Tiederman
,
W. G.
, 1975, “
Laser-Doppler Anemometer Measurements in Drag-Reducing Channel Flows
,”
J. Fluid Mech.
0022-1120,
70
(
2
), pp.
369
392
.
39.
Virk
,
P. S.
, 1971, “
Drag Reductions in Rough Pipes
,”
J. Fluid Mech.
0022-1120,
45
, pp.
225
246
.
40.
Edwards
,
M. F.
, and
Smith
,
R.
, 1980, “
The Use of Eddy Viscosity Expressions for Predicting Velocity Profiles in Newtonian, Non-Newtonian and Drag-Reducing Turbulent Pipe Flow
,”
J. Non-Newtonian Fluid Mech.
0377-0257,
7
, pp.
153
169
.
41.
Gust
,
G.
, 1976, “
Observations on Turbulent=Drag Reduction in a Dilute Suspension of Clay in Sea-Water
,”
J. Fluid Mech.
0022-1120,
75
(
1
), pp.
29
47
.
42.
Virk
,
P. S.
,
Merrill
,
E. W.
,
Mickley
,
H. S.
, and
Smith
,
K. A.
, 1967, “
The Toms Phenomenon: Turbulent Pipe Flow of Dilute Polymer Solutions
,”
J. Fluid Mech.
0022-1120,
30
(
2
), pp.
305
328
.
43.
Yang
,
S. Q.
, and
Dou
,
G.
, 2008, “
Modeling of Viscoelastic Turbulent Flow in Open Channel and Pipe
,”
Phys. Fluids
1070-6631,
20
(
6
), p.
065105
.
You do not currently have access to this content.