This study makes an attempt to investigate Newtonian/non-Newtonian pipe flows in a laminar-turbulent transition region, which is an extraordinarily complicated process and is not fully understood. The key characteristic of this region is its intermittent nature, i.e., the flow alternates in time between being laminar or turbulent in a certain range of Reynolds numbers. The physical nature of this intermittent flow can be aptly described with the aid of the intermittency factor γ, which is defined as that fraction of time during which the flow at a given position remains turbulent. Spriggs postulated that a weighting factor can be used to calculate the friction factor, applying its values in laminar and turbulent states. Based on these, a model is developed to empirically express the mean velocity and Reynolds shear stress in the transition region. It is found that the intermittency factor can be used as a weighting factor for calculating the flow structures in the transition region. Good agreements can be achieved between the calculations and experimental data available in the literature, indicating that the present model is acceptable to express the flow characteristics in the transition region.

References

References
1.
Reynolds
,
O.
,
1883
, “
An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law of Resistance in Parallel Channels
,”
Proc. R. Soc. London
,
35
(
224–226
), pp.
84
99
.10.1098/rspl.1883.0018
2.
Schlichting
,
H.
,
1979
,
Boundary Layer Theory
,
McGraw-Hill
,
New York
, p.
452
.
3.
Wygnanski
,
I.
, and
Champagne
,
F.
,
1973
, “
On Transition in a Pipe. Part 1. The Origin of Puffs and Slugs and the Flow in a Turbulent Slug
,”
J. Fluid Mech.
,
59
(
02
), pp.
281
335
.10.1017/S0022112073001576
4.
Wygnanski
,
I. J.
,
Sokolov
,
M.
, and
Friedman
,
D.
,
1975
, “
On Transition in a Pipe. Part 2. The Equilibrium Puff
,”
J. Fluid Mech.
,
69
(
02
), pp.
283
304
.10.1017/S0022112075001449
5.
Darbyshire
,
A.
, and
Mullin
,
T.
,
1995
, “
Transition to Turbulence in Constant-Mass-Flux Pipe Flow
,”
J. Fluid Mech.
,
289
, pp.
83
114
.10.1017/S0022112095001248
6.
Draad
,
A. A.
,
Kuiken
,
G.
, and
Nieuwstadt
,
F.
,
1998
, “
Laminar–Turbulent Transition in Pipe Flow for Newtonian and Non-Newtonian Fluids
,”
J. Fluid Mech.
,
377
, pp.
267
312
.10.1017/S0022112098003139
7.
Eliahou
,
S.
,
Tumin
,
A.
, and
Wygnanski
,
I.
,
1998
, “
Laminar–Turbulent Transition in Poiseuille Pipe Flow Subjected to Periodic Perturbation Emanating From the Wall
,”
J. Fluid Mech.
,
361
, pp.
333
349
.10.1017/S002211209800888X
8.
Hof
,
B.
,
Juel
,
A.
, and
Mullin
,
T.
,
2003
, “
Scaling of the Turbulence Transition Threshold in a Pipe
,”
Phys. Rev. Lett.
,
91
(
24
), p.
244502
.10.1103/PhysRevLett.91.244502
9.
Han
,
G.
,
Tumin
,
A.
, and
Wygnanski
,
I.
,
2000
, “
Laminar-Turbulent Transition in Poiseuille Pipe Flow Subjected to Periodic Perturbation Emanating From the Wall. Part 2. Late Stage of Transition
,”
J. Fluid Mech.
,
419
, pp.
1
27
.10.1017/S0022112000001269
10.
Ben-Dov
,
G.
, and
Cohen
,
J.
,
2007
, “
Critical Reynolds Number for a Natural Transition to Turbulence in Pipe Flows
,”
Phys. Rev. Lett.
,
98
(
6
), p.
064503
.10.1103/PhysRevLett.98.064503
11.
Hof
,
B.
,
van Doorne
,
C. W.
,
Westerweel
,
J.
,
Nieuwstadt
,
F. T.
,
Faisst
,
H.
,
Eckhardt
,
B.
,
Wedin
,
H.
,
Kerswell
,
R. R.
, and
Waleffe
,
F.
,
2004
, “
Experimental Observation of Nonlinear Traveling Waves in Turbulent Pipe Flow
,”
Science
,
305
(
5690
), pp.
1594
1598
.10.1126/science.1100393
12.
Matas
,
J.-P.
,
Morris
,
J. F.
, and
Guazzelli
,
E.
,
2003
, “
Transition to Turbulence in Particulate Pipe Flow
,”
Phys. Rev. Lett.
,
90
(
1
), p.
014501
.10.1103/PhysRevLett.90.014501
13.
Keefe
,
L.
,
Moin
,
P.
, and
Kim
,
J.
,
1992
, “
The Dimension of Attractors Underlying Periodic Turbulent Poiseuille Flow
,”
J. Fluid Mech.
,
242
, pp.
1
29
.10.1017/S0022112092002258
14.
Karami
,
H.
, and
Mowla
,
D.
,
2012
, “
Investigation of the Effects of Various Parameters on Pressure Drop Reduction in Crude Oil Pipelines by Drag Reducing Agents
,”
J. Non-Newton. Fluid Mech.
,
177
, pp.
37
45
.10.1016/j.jnnfm.2012.04.001
15.
Rotta
J.
,
1956
, “
Experimenteller Beitrag zur Entstehung Turbulenter Strömung im Rohr
,”
Arch. Appl. Mech.
,
24
(
4
), pp.
258
281
.
16.
Toms
,
B. A.
,
1948
, “
Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers
,”
Proceedings of the 1st International Congress on Rheology
, pp.
135
141
.
17.
Virk
,
P.
,
1971
, “
Drag Reduction in Rough Pipes
,”
J. Fluid Mech.
,
45
(
02
), pp.
225
246
.10.1017/S0022112071000028
18.
Virk
,
P.
,
1971
, “
An Elastic Sublayer Model for Drag Reduction by Dilute Solutions of Linear Macromolecules
,”
J. Fluid Mech.
,
45
(
03
), pp.
417
440
.10.1017/S0022112071000120
19.
Virk
,
P. S.
,
1975
, “
Drag Reduction Fundamentals
,”
AIChE J.
,
21
(
4
), pp.
625
656
.10.1002/aic.690210402
20.
Virk
,
P. S.
,
Merrill
,
E.
,
Mickley
,
H.
,
Smith
,
K.
, and
Mollo-Christensen
,
E.
,
1967
, “
The Toms Phenomenon: Turbulent Pipe Flow of Dilute Polymer Solutions
,”
J. Fluid Mech.
,
30
(
02
), pp.
305
328
.10.1017/S0022112067001442
21.
Warholic
,
M.
,
Massah
,
H.
, and
Hanratty
,
T.
,
1999
, “
Influence of Drag-Reducing Polymers on Turbulence: Effects of Reynolds Number, Concentration and Mixing
,”
Exp. Fluids
,
27
(
5
), pp.
461
472
.10.1007/s003480050371
22.
Den Toonder
,
J.
,
Hulsen
,
M.
,
Kuiken
,
G.
, and
Nieuwstadt
,
F.
,
1997
, “
Drag Reduction by Polymer Additives in a Turbulent Pipe Flow: Numerical and Laboratory Experiments
,”
J. Fluid Mech.
,
337
, pp.
193
231
.10.1017/S0022112097004850
23.
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.
,
486
, pp.
213
238
.10.1017/S0022112003004610
24.
Ptasinski
,
P.
,
Nieuwstadt
,
F.
,
Van Den Brule
,
B.
, and
Hulsen
,
M.
,
2001
, “
Experiments in Turbulent Pipe Flow With Polymer Additives at Maximum Drag Reduction
,”
Flow, Turbul. Combust.
,
66
(
2
), pp.
159
182
.10.1023/A:1017985826227
25.
Roy
,
A.
, and
Larson
,
R. G.
,
2005
, “
A Mean Flow Model for Polymer and Fiber Turbulent Drag Reduction
,”
Appl. Rheol.
,
15
(
6
), pp.
370
389
.10.3933/ApplRheol-15-370
26.
White
,
C.
,
Somandepalli
,
V.
, and
Mungal
,
M.
,
2004
, “
The Turbulence Structure of Drag-Reduced Boundary Layer Flow
,”
Exp. Fluids
,
36
(
1
), pp.
62
69
.10.1007/s00348-003-0630-0
27.
Esmael
,
A.
,
Nouar
,
C.
,
Lefèvre
,
A.
, and
Kabouya
,
N.
,
2010
, “
Transitional Flow of a Non-Newtonian Fluid in a Pipe: Experimental Evidence of Weak Turbulence Induced by Shear-Thinning Behavior
,”
Phys. Fluids
,
22
, p. 101701.10.1063/1.3491511
28.
White
,
C. M.
, and
Mungal
,
M. G.
,
2008
, “
Mechanics and Prediction of Turbulent Drag Reduction With Polymer Additives
,”
Annu. Rev. Fluid Mech.
,
40
, pp.
235
256
.10.1146/annurev.fluid.40.111406.102156
29.
Yang
,
S.-Q.
, and
Dou
,
G.
,
2005
, “
Drag Reduction in a Flat-Plate Boundary Layer Flow by Polymer Additives
,”
Phys. Fluids
,
17
(
6
), p.
065104
.10.1063/1.1924650
30.
Yang
,
S.-Q.
, and
Dou
,
G.-R.
,
2008
, “
Modeling of Viscoelastic Turbulent Flow in Channel and Pipe
,”
Phys. Fluids
,
20
(
6
), p.
065105
.10.1063/1.2920275
31.
Yang
,
S.-Q.
,
2009
, “
Drag Reduction in Turbulent Flow With Polymer Additives
,”
ASME J. Fluids Eng.
,
131
(
5
), p.
051301
.10.1115/1.3111255
32.
Yang
,
S.-Q.
, and
Dou
,
G.
,
2010
, “
Turbulent Drag Reduction With Polymer Additive in Rough Pipes
,”
J. Fluid Mech.
,
642
, pp.
279
294
.10.1017/S002211200999187X
33.
Yang
,
S.-Q.
, and
Ding
,
D.
,
2013
, “
Drag Reduction Induced by Polymer in Turbulent Pipe Flows
,”
Chem. Eng. Sci.
,
102
, pp.
200–208
.10.1016/j.ces.2013.07.048
34.
L'vov
,
V. S.
,
Pomyalov
,
A.
,
Procaccia
,
I.
, and
Tiberkevich
,
V.
,
2004
, “
Drag Reduction by Polymers in Wall Bounded Turbulence
,”
Phys. Rev. Lett.
,
92
(
24
), p.
244503
.10.1103/PhysRevLett.92.244503
35.
Emmons
,
H. W.
,
2012
, “
The Laminar-Turbulent Transition in a Boundary Layer-Part I
,”
J. Aeronaut. Sci.
,
18
(
7
), pp.
121–131
.
36.
Dou
,
G.
,
1996
, “
Basic Law in Mechanics of Turbulent Flows
,”
China Ocean Eng.
,
10
, pp.
1
45
.
37.
Spriggs
,
H. D.
,
1973
, “
Comments on Transition From Laminar to Turbulent Flow
,”
Ind. Eng. Chem. Fund.
,
12
(
3
), pp.
286
290
.10.1021/i160047a004
38.
Lumley
,
J. L.
,
1969
, “
Drag Reduction by Additives
,”
Annu. Rev. Fluid Mech.
,
1
(
1
), pp.
367
384
.10.1146/annurev.fl.01.010169.002055
39.
Yang
,
S.-Q.
,
Lim
,
S.-Y.
, and
McCorquodale
,
J.
,
2005
, “
Investigation of Near Wall Velocity in 3-D Smooth Channel Flows
,”
J. Hydraul. Res.
,
43
(
2
), pp.
149
157
.10.1080/00221686.2005.9641231
40.
Yang
,
S.-Q.
,
2010
,
“Conditionally Averaged Turbulent Structures in 2D Channel Flow
,”
Proc. ICE—Water Manage.
,
163
(
2
), pp.
79
88
.10.1680/wama.2010.163.2.79
41.
Presti
,
F.
,
2000
, “
Investigation of Transitional and Turbulent Pipe Flow of Non-Newtonian Fluids
,”
Ph.D. thesis, University of Liverpool
, Liverpool, UK.
42.
Escudier
,
M.
,
Presti
,
F.
, and
Smith
,
S.
,
1998
, “
Drag Reduction in the Turbulent Pipe Flow of Polymers
,”
J. Non-Newton. Fluid Mech.
,
81
(
3
), pp.
197
213
.10.1016/S0377-0257(98)00098-6
43.
Narayanan
,
M. A. B.
, and
Narayana
,
T.
,
1967
, “
Some Studies on Transition From Laminar to Turbulent Flow in a Two-Dimensional Channel
,”
ZAMP
,
18
(
5
), pp.
642
650
.10.1007/BF01602037
44.
Rao
,
G.
,
1974
, “
Mechanics of Transition in an Axisymmetric Laminar Boundary Layer on a Circular Cylinder
,”
ZAMP
,
25
(
1
), pp.
63
75
.10.1007/BF01602109
45.
Narasimha
,
R.
,
1985
, “
The Laminar-Turbulent Transition Zone in the Boundary Layer
,”
Prog. Aerospace Sci.
,
22
(
1
), pp.
29
80
.10.1016/0376-0421(85)90004-1
46.
Wójs
,
K.
,
1993
, “
Laminar and Turbulent Flow of Dilute Polymer Solutions in Smooth and Rough Pipes
,”
J. Non-Newton. Fluid Mech.
,
48
(
3
), pp.
337
355
.10.1016/0377-0257(93)87027-M
47.
Peixinho
,
J.
,
Nouar
,
C.
,
Desaubry
,
C.
, and
Théron
,
B.
,
2005
, “
Laminar Transitional and Turbulent Flow of Yield Stress Fluid in a Pipe
,”
J. Non-Newton. Fluid Mech.
,
128
(
2
), pp.
172
184
.10.1016/j.jnnfm.2005.03.008
48.
Dou
,
G.
, and
Wang
,
G.
,
1988
, “
The Turbulence Structure of a Viscoelastic Fluid by Dilute Solution of Linear Macromolecules in the Interim State From Laminar Flow to Turbulent Flow
,”
J. Nanjing Hydraul. Res.
,
3
, pp.
1
14
.
49.
Jiménez
,
J.
, and
Pinelli
,
A.
,
1999
, “
The Autonomous Cycle of Near-Wall Turbulence
,”
J. Fluid Mech.
,
389
, pp.
335
359
.10.1017/S0022112099005066
You do not currently have access to this content.