The friction factor data from transitional rough test pipes, from the measurements of Sletfjerding and Gudmundsson (2003, “Friction Factor Directly From Roughness Measurements,” J. Energy Resour. Technol. 125, pp. 126–130), have been analyzed in terms of directly measurable roughness parameters, Ra the arithmetic mean roughness, RZ the mean peak to valley heights roughness, Rq the root mean square (rms) roughness, and RqH rms textured roughness (H, the Hurst exponent is a texture parameter), in addition to h the equivalent sand grain roughness. The proposed friction factor λ, in terms of new scaling parameter, viz., the roughness Reynolds number Reϕ=Reϕ (where ϕ is a nondimensional roughness scale), is a universal relation for all kinds of surface roughness. This means that Prandtl’s smooth pipe friction factor relation would suffice provided that the traditional Reynolds number Re is replaced by the roughness Reynolds number Reϕ. This universality is very well supported by the extensive rough pipe data of Sletfjerding and Gudmundsson, Shockling’s (2005, “Turbulence Flow in Rough Pipe,” MS thesis, Princeton University) machined honed pipe surface roughness data, and Nikuradse’s (1933, Laws of Flow in Rough Pipe, VDI, Forchungsheft No. 361) sand grain roughness data. The predictions for the roughness function ΔU+, and the roughness scale ϕ for inflectional roughness compare very well with the data of the above mentioned researchers. When surface roughness is present, there is no universality of scaling of the friction factor λ with respect to the traditional Reynolds number Re, and different expressions are needed for various types of roughnesses, as suggested, for example, with inflectional roughness, monotonic roughness, etc. In traditional variables, the proposed friction factor prediction for inflectional roughness in the pipes, is supported very well by the experimental data of Sletfjerding and Gudmundsson, Shockling, and Nikuradse. In the present work, the predictions of friction factor as implicit relations, as well as approximate explicit relations, have also been proposed for various roughness scales.

1.
Nikuradse
,
J.
, 1933,
Laws of Flow in Rough Pipe
,
VDI
, Forchungsheft No. 361, English translation NACA T.M. 1292, 1950.
2.
Colebrook
,
C. F.
, 1939, “
Turbulent Flow in Pipes With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws
,”
Journal of Institution of Civil Engineers
, London,
11
, pp.
133
156
.
3.
Sletfjerding
,
E.
, 1999, “
Friction Factor in Smooth and Rough Gas Pipelines
,” Dr.-Ing. thesis, Norwegian University of Science and Technology, Trondheim.
4.
Sletfjerding
,
E.
, and
Gudmundsson
,
J. S.
, 2003, “
Friction Factor Directly From Roughness Measurements
,”
J. Energy Resour. Technol.
0195-0738,
125
, pp.
126
130
.
5.
Barr
,
D. I. H.
, 1972, “
New Forms of Equations for the Correlation of Pipe Resistance Data
,”
Proc. Inst. Civil Engg. London
,
53
(
2
), pp.
383
390
;
see also 1977, “
Discussion of Accurate Explicit Equation for Friction Factor
,”
J. Hydr. Div.
0044-796X ASCE,
103
(
HY3
), pp.
334
337
.
6.
Churchill
,
S. W.
, 1973, “
Empirical Expressions for the Shear Stress in Turbulent Flow in Commercial Pipe
,”
AIChE J.
0001-1541,
19
, pp.
375
376
.
7.
Swamee
,
P. K.
, and
Jain
,
A. K.
, 1976, “
Explicit Equations for Pipe Flow Problems
,”
J. Hydraul. Eng.
ASCE,
102
(
5
), pp.
657
664
;
Collins
,
M. A.
, 1976,
J. Hydraul. Eng.
102
(
11
), pp.
1707
1709
;
Barr
,
D. H.
, 1977,
J. Hydraul. Eng.
103
(
4
), pp.
460
463
.
8.
Haaland
,
S.
, 1983, “
Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow
,”
J. Fluids Eng.
0098-2202,
105
, pp.
89
90
.
9.
Prandtl
,
L.
, 1935, “
The Mechanics of Viscous Fluids
,” in
Aerodynamic Theory
, edited by
Durand
,
W. F.
,
California Institute of Technology
,
Pasadena, CA
, Vol.
3
, pp.
34
208
.
10.
McKeon
,
B. J.
, 2003, “
High Reynolds Number Turbulent Pipe Flow
,” Ph.D. thesis, Princeton University, Princeton.
11.
Warburton
,
C.
, 1974, “
Surface Roughness of Graphite and Its Effect on Friction Factor
,”
Proc. Inst. Mech. Eng.
0020-3483,
188
, pp.
457
460
.
12.
Clauser
,
F. H.
, 1954, “
Turbulent Boundary Layers in Adverse Pressure Gradients
,”
J. Aeronaut. Sci.
0095-9812,
21
, pp.
91
108
.
13.
Hama
,
F. R.
, 1954, “
Boundary-Layer Characteristics for Rough and Smooth Surfaces
,”
Soc. Nav. Archit. Mar. Eng., Trans.
0081-1661,
62
, pp.
333
351
.
14.
Afzal
,
N.
,
Seena
,
A.
, and
Bushra
,
A.
, 2006, “
Power Law Turbulent Velocity Profile in Transitional Rough Pipes
,”
J. Fluids Eng.
0098-2202,
128
(
3
), pp.
548
558
.
15.
Raupach
,
M. R.
,
Antonia
,
R. A.
, and
Rajagopalan
,
S.
, 1991, “
Rough-Wall Turbulent Boundary Layer
,”
Adv. Appl. Mech.
0065-2156,
44
, pp.
1
25
.
16.
Jimenez
,
J.
, 2004, “
Turbulent Flow Over Rough Walls
,”
Annu. Rev. Fluid Mech.
0066-4189,
36
, pp.
173
196
.
17.
Bendrict
,
R.
, 1980,
Fundamentals of Pipe Flow
,
Wiley
,
New York
, p.
240
.
18.
Balachandar
,
R.
,
Hager
,
K.
, and
Blackely
,
D.
, 2002, “
Velocity Distribution in Decelerating Flow Over a Surface
,”
Can. J. Civ. Eng.
0315-1468,
29
, pp.
211
221
.
19.
Hager
,
W. H.
, 2003, “
Blasius: A Life in Research and Education
,”
Exp. Fluids
0723-4864,
34
, pp.
568
571
.
20.
Nikuradse
,
J.
, 1932,
Laws of Turbulent Flow in Smooth Pipes
,
VDI
, Forchungsheft No. 356, English translation NASA TT F-10, 1966.
21.
Afzal
,
N.
, 1982, “
Fully Developed Turbulent Flow in a Pipe: An Intermediate Layer
,”
Arch. Appl. Mech.
0939-1533,
52
, pp.
355
377
.
22.
Afzal
,
N.
, 1984, “
The Mesolayer Theory of Turbulent Flows
,”
AIAA J.
0001-1452,
22
, pp.
437
439
.
23.
Afzal
,
N.
, and
Seena
,
A.
, 2007, “
Alternate Scales for Turbulent Flow in Transitional Rough Pipes: Universal Log Law
,”
J. Fluids Eng.
0098-2202,
129
(
1
), pp.
80
90
.
24.
Moody
,
L. F.
, 1944, “
Friction Factors for Pipe Flow
,”
Trans. ASME
0097-6822,
66
, pp.
671
684
.
25.
Yen
,
B. C.
, 2002, “
Open Channel Flow Resistance
,”
J. Hydraul. Eng.
0733-9429,
128
, pp.
20
39
.
26.
Grigson
,
C.
, 1987, “
The Full Scale Drag of Ship Surface and the Effects of Quality Roughness on Predicted Power
,”
J. Ship Res.
0022-4502,
31
, pp.
189
206
.
27.
Cebeci
,
T.
, 2004,
Analysis of Turbulent Flows
,
Elsevier
,
New York
.
28.
Shockling
,
M. A.
, 2005, “
Turbulent Flow in Rough Pipe
,” MSE thesis, Princeton University, Princeton.
29.
Shockling
,
M. A.
,
Allen
,
J. J.
, and
Smits
,
A. J.
, 2006, “
Roughness Effects in Turbulent Pipe Flow
,”
J. Fluid Mech.
0022-1120,
564
, pp.
267
285
.
30.
Afzal
,
N.
, 1976, “
Millikan Argument at Moderately Large Reynolds Numbers
,”
Phys. Fluids
0031-9171,
19
, pp.
600
602
.
31.
Izakson
,
A. A.
, 1937, “
On Formula for the Velocity Distribution Near Walls
,”
Zh. Tekh. Fiz.
0044-4642,
4
, pp.
155
159
.
32.
Millikan
,
C. B.
, 1938, “
A Critical Discussion of Turbulent Flows in Channels and Circular Tubes
,”
Proceedings of the Fifth International Congress on Applied Mechanics
, edited by
J. P.
Den Hartog
, and
H.
Peters
,
Wiley
,
New York
, pp.
386
392
.
33.
Patel
,
V. C.
, and
Head
,
M. R.
, 1969, “
Some Observations on Skin Friction and Velocity Profile in Fully Developed Pipe and Channel Flow
,”
J. Fluid Mech.
0022-1120,
38
, pp.
181
201
.
34.
Afzal
,
N.
, and
Yajnik
,
K.
, 1973, “
Analysis of Turbulent Pipe and Channel Flows at Moderately Large Reynolds Number
,”
J. Fluid Mech.
0022-1120,
61
, pp.
23
31
.
35.
Zagarola
,
M. V.
, and
Smits
,
A. J.
, 1998, “
Mean Flow Scaling in Turbulent Pipe Flow
,”
J. Fluid Mech.
0022-1120,
373
, pp.
33
79
.
36.
McKeon
,
B. J.
,
Zagarola
,
M. V.
, and
Smits
,
A. J.
, 2005, “
A New Friction Factor Relationship for Fully Developed Pipe Flow
,”
J. Fluid Mech.
0022-1120,
538
, pp.
429
443
.
37.
Spalding
,
D. B.
, 1961, “
A Single Formula for the Law of the Wall
,”
J. Appl. Mech.
0021-8936,
28
, pp.
455
458
.
38.
McKeon
,
B. J.
,
Swanson
,
C. J.
,
Zagarola
,
M. V.
,
Donnelly
,
R. J.
, and
Smits
,
A. J.
, 2004, “
Friction Factor for Smooth Pipe Flow
,”
J. Fluid Mech.
0022-1120,
511
, pp.
41
44
.
39.
Allen
,
J. J.
,
Shockling
,
M. A.
, and
Smits
,
A. J.
, 2005, “
Evaluation of a Universal Transitional Resistance Diagram for Pipes With Honed Surfaces
,”
Phys. Fluids
1070-6631,
17
, p.
121702
.
You do not currently have access to this content.