Alternate power law velocity profile u+=Aζα in transitional rough pipe fully turbulent flow, has been proposed, in terms of new appropriate inner rough wall variables (ζ=Z+ϕ, uϕ=uϕ), and new parameters Rϕ=Rτϕ termed as the roughness friction Reynolds number, Reϕ=Reϕ termed as the roughness Reynolds number and ϕ termed as roughness scale (along with normal wall coordinate Z=y+ϵr where ϵr is the shift of the origin of boundary layer due to the rough wall, Z+=Zuτν and u+=uuτ). The envelope of the power law shows that the power law constants α and A depend on the parameter Rϕ (i.e., α=α(Rϕ) and A=A(Rϕ)) but explicitly independent of the wall roughness parameter hδ (roughness height h in pipe of radius δ). The roughness scale ϕ has been related to the roughness function ΔU+ of Clauser representing the velocity shift caused by wall roughness. The present results of the velocity profile, just slightly above the wall roughness level h, remain valid for all types of wall roughness. The data of Nikuradse for sand-grain roughness, in transitional and fully rough pipes, has been considered, which provides good support to the predictions of an alternate power law velocity profile, based on single parameter Rϕ, the roughness friction Reynolds number.

1.
Nikuradse
,
J.
, 1932, “
Laws of Turbulent Flow in Smooth Pipes
,” VDI, Forchungsheft N-356 (English translation NACA TTF-10, p.
359
)
2.
Barenblatt
,
G. I.
, 1993, “
Scaling Laws for Fully Developed Turbulent Shear Flows, Part I: Basic Hypothesis and Analysis
,”
J. Fluid Mech.
0022-1120,
248
, pp.
513
520
.
3.
Barenblatt
,
G. I.
,
Chorin
,
A. J.
, and
Prostokishin
,
V. M.
, 1997, “
Scaling Laws for Fully Developed Turbulent Flow in Pipes
,”
Appl. Mech. Rev.
0003-6900,
90
, pp.
413
429
.
4.
Kailasnath
,
P.
, 1993, “
Reynolds Number Effect and The Momentum Flux in Turbulent Boundary Layer
,” Ph.D. thesis, Mason Lab., Yale University.
5.
Zagarola
,
M. V.
,
Perry
,
A. E.
, and
Smits
,
A. J.
, 1997, “
Log Laws or Power Laws: The Scaling in the Overlap Region
,”
Phys. Fluids
1070-6631,
9
, pp.
2094
2100
.
6.
Afzal
,
N.
, 2001, “
Power Law and Log Law Velocity Profiles in Fully Developed Turbulent Pipe Flow: Equivalent Relations at Large Reynolds Numbers
,”
Acta Mech.
0001-5970,
151
, pp.
171
183
.
7.
George
,
W.
, and
Castillo
,
L.
, 1997, “
Zero Pressure Gradient Turbulent Boundary Layer
,”
Appl. Mech. Rev.
0003-6900,
50
, pp.
689
729
.
8.
Afzal
,
N.
, 1997, “
Power Law in Wall and Wake Layers of a Turbulent Boundary Layer
,”
Proceedings Seventh Asian Congress of Fluid Mechanics
,
Allied Publishers Limited
, New Delhi, pp.
805
808
.
9.
Afzal
,
N.
, 2001, “
Power Law and Log Law Velocity Profiles in Turbulent Boundary Layer: Equivalent Relations at Large Reynolds Numbers
,”
Acta Mech.
0001-5970,
151
, pp.
195
216
.
10.
Afzal
,
N.
, 2005, “
Scaling of Power Law Velocity Profile in Wall-Bounded Turbulent Shear Flows.
,” AIAA-2005-0109, 43rd AIAA Aerospace Sciences Meeting and Exhibit, 10–13 Jan., Reno, Nevada.
11.
Panton
,
R. L.
, 2002, “
Evaluation of the Barenblatt-Chorin-Prostokishin Power Law for Turbulent Boundary Layers
,”
Phys. Fluids
1070-6631,
14
, pp.
1806
1808
.
12.
Yaglom
,
A. M.
, 2001, “
The Century of Turbulence Theory: The Main Achievements and Unsolved Problems
,”
New Trends in Turbulence: Nouveaux Aspects
,
M.
Lesieur
,
A.
Yaglom
, and
F.
David
, eds.,
Les Houches: EPT
,
Springer-Verlag
, Berlin, Vol.
74
.
13.
Prandtl
,
L.
, 1934, “
The Mechanics of Viscous Fluids
,” in
Aerodynamics Theory, Vol. 3
,
W. F.
Durand
, ed.,
California Institute of Technology
,
Pasadena, California
, pp.
34
208
.
14.
Buschmann
,
M. H.
, and
Gad-el-Hak
,
K.
, 2003, “
The Debate Concerning the Mean Velocity Profile of a Turbulent Boundary Layer
,”
AIAA J.
0001-1452,
41
(
4
), pp.
565
572
.
15.
Buschmann
,
M. H.
, and
Gad-el-Hak
,
K.
, 2003, “
Generalized Logarithmic Law and its Consequences
,”
AIAA J.
0001-1452,
41
(
1
), pp.
40
48
.
16.
Eaton
,
J. K.
, and
Nagib
,
H. M.
, 2004, “
Report: ‘Second International Workshop on Wall-Bounded Turbulent Flows
,’ by H. Nagib and A. J. Smits,” from 2–5 November, the Abdus Salem International Center for Theoretical Physics, Trieste, Italy.
17.
Afzal
,
N.
, 2005, “
Analysis of Power Law and Log Law Velocity Profiles in Overlap Region of a Turbulent Wall Jet
,”
Proc. R. Soc. London, Ser. A
1364-5021,
461
, pp.
1889
1910
.
18.
Wosnik
,
M.
,
Castillo
,
L.
, and
George
,
W. K.
, 2000, “
A Theory for Turbulent Pipe and Channel Flows
,”
J. Fluid Mech.
0022-1120,
421
, pp.
115
145
.
19.
Schultz
,
M. P.
, and
Myers
,
A.
, 2003, “
Comparison of Three Roughness Function Determination Methods
,”
Exp. Fluids
0723-4864,
35
(
4
), pp.
372
379
.
20.
Millikan
,
C. B.
, 1938, “
A Critical Discussion of Turbulent Flow in Channels and Circular tubes
,”
Proc. Fifth. International Congress of Applied Mechaics
,
J. P.
Den Hartog
and
H.
Peters
, eds.,
Wiley
, New York, pp.
386
392
.
21.
Schlichting
,
H.
, 1968,
Boundary Layer Theory
,
McGraw-Hill
, New York.
22.
Raupach
,
M. R.
,
Antonia
,
R. A.
, and
Rajagopalan
,
S.
, 1991, “
Rough-Wall Turbulent Boundary Layer
,”
Adv. Appl. Mech.
0065-2156,
44
, pp.
1
25
.
23.
Piquet
,
J.
, 1999,
Turbulent Flow
,
Springer-Verlag
, Berlin.
24.
Jimenez
,
J.
, 2004, “
Turbulent Flow Over Rough Walls
,”
Annu. Rev. Fluid Mech.
0066-4189,
36
, pp.
173
196
.
25.
Nikuradse
,
J.
, 1933, “
Laws of Flow in Rough Pipe
,” VDI, Forchungsheft N-361 (English translation NACA TM 1292, 1950).
26.
Porporato
,
A.
, and
Sordo
,
S.
, 2001, “
On the Incomplete Similarity for Turbulent Velocity Profiles in Rough Pipes
,”
Phys. Fluids
1070-6631,
13
(
9
), pp.
2596
2601
.
27.
Balachandar
,
R.
,
Hagel
,
K.
, and
Blakely
,
D.
, 2002, “
Velocity Distribution in Decelerating Flow Over Rough Surface
”,
Can. J. Civ. Eng.
0315-1468,
29
,
211
221
.
28.
Balachandar
,
R.
,
Blackely
,
D.
, and
Bugg
,
J.
, 2002, “
Friction Factor and the Power Law Velocity Profile in Smooth and Rough Shallow Open Channel Flow
Can. J. Civ. Eng.
0315-1468,
29
, pp.
256
266
.
29.
Clauser
,
F. H.
, 1954, “
Turbulent Boundary Layers in Adverse Pressure Gradients
,”
J. Aeronaut. Sci.
0095-9812,
4
,
91
108
.
30.
Hama
,
F. R.
, 1954, “
Boundary Layer Characteristics for Smooth and Rough Surfaces
,”
Soc. Nav. Archit. Mar. Eng., Trans.
0081-1661,
63
, pp.
353
358
.
31.
Seo
,
J.
, 2003, “
Investigation of the Upstream Conditions and Surface Roughness in Turbulent Boundary Layer
,” Ph.D. thesis, Rensselaer Polytechnic Institute, New York.
32.
Seo
,
J.
, and
Castillo
,
L.
, 2004, “
Rough Surface Turbulent Boundary Layer: The Composite Profiles
,” AIAA 2004-1287, 42nd AIAA Aerospace Sciences Meeting and Exhibit, 5–8 Jan., Reno, Nevada.
33.
Kotey
,
N. A.
,
Bergstrom
,
D. J.
, and
Tachie
,
T. F.
, 2003, “
Power Law for Rough Wall Turbulent Boundary Layer
,”
Phys. Fluids
1070-6631,
15
,
1396
1404
.
34.
Flack
,
K. A
,
Schultz
,
M. P.
, and
Shapiro
,
T. A.
, 2005, “
Experimental Support for Townsend’s Reynolds Number Similarity Hypothesis on Rough Walls
,”
Phys. Fluids
1070-6631,
17
, p.
035102
.
35.
George
,
W.
,
Castillo
,
L.
, and
Knecht
,
P.
, 1996, “
The Zero Pressure Gradient Turbulent Boundary Layer
,” Technical Report TRL-153 Turbulence Research Lab., State University of New York at Buffalo.
36.
Colebrook
,
C. F.
, 1939, “
Turbulent Flow in Pipes With particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws
,”
Proc. Inst. of Civ. Eng. (UK)
0307-8353,
11
, pp.
133
156
.
37.
Moody
,
L. F.
, 1944, “
Friction Factor for Pipe Flow
,”
Trans. ASME
0097-6822,
66
, pp.
676
684
.
38.
Zagarola
,
M. V.
, and
Smits
,
A. J.
, 1998, “
Mean Flow Scaling in Turbulent Pipe Flow
,”
J. Fluid Mech.
0022-1120,
373
,
33
79
.
39.
Smith
,
D. W.
, and
Walker
,
J. H.
, 1958, “
Skin Friction Measurements in Incompressible Flow
,” NACA TN 4231.
40.
Zanoun
,
E. S.
,
Durst
,
F.
, and
Nagib
,
H.
, 2003, “
Evaluating the Law of the Wall in Two Dimensional Fully Developed Turbulent Channel Flow
,”
Phys. Fluids
1070-6631,
15
, pp.
3079
3089
.
41.
Tennekes
,
H.
, 1968, “
Outline of a Second Order Theory of Turbulent Pipe Flow
,”
AIAA J.
0001-1452,
6
, pp.
1735
1740
.
42.
Afzal
,
N.
, and
Yajnik
,
K.
, 1973, “
Analysis of Turbulent Pipe and Channel Flow at Moderately large Reynolds number
,”
J. Fluid Mech.
0022-1120,
61
, pp.
23
31
.
43.
Afzal
,
N.
, 1976, “
Millikan’s Argument at Moderately Large Reynolds Numbers
,”
Phys. Fluids
0031-9171,
19
, pp.
600
602
.
44.
Antonia
,
R. A.
, and
Krogstad
,
P.-A.
, 2001, “
Turbulence Structure in Boundary Layer Over Different types of Surface Roughness
,”
Fluid Dyn. Res.
0169-5983,
28
, pp.
139
157
.
45.
Afzal
,
N.
, 2005, “
Power Law Velocity Profile in a Turbulent Boundary Layer on Transitional Rough Walls
,” in preparation.
46.
Guo
,
J.
, 2001, “
Discussion on ‘A Simple Method of Measuring Shear Stress on Rough Boundaries
,'”
J. Hydraul. Res.
0022-1686,
39
, pp.
445
226
.
47.
Cebeci
,
T.
, 2004,
Analysis of Turbulent Flows
,
Elsevier
, New York.
48.
Afzal
,
N.
, and
Seena
,
A.
, 2005, “
Alternate Scales for Turbulent Flow in Transitional Rough Pipes: Universal Log laws
,” in prepartion.
49.
Afzal
,
N.
, 2005, “
Alternate Scales for Turbulent Boundary Layers on Transitional Rough Walls: Universal Log Laws
,” in preparation.
This content is only available via PDF.
You do not currently have access to this content.