0
Journal of Fluids Engineering | Volume 128 | Issue 3 | TECHNICAL PAPERS
TECHNICAL PAPERS

Power Law Turbulent Velocity Profile in Transitional Rough Pipes

[+] Author and Article Information
Noor Afzal, Abu Seena

Department of Mechanical Engineering, Aligarh University, Aligarh 202 002, India

Afzal Bushra

Department of Civil and Environment Engineering, University of Windsor, Windsor, Ontario N9B 3P4, Canada

J. Fluids Eng 128(3), 548-558 (Oct 02, 2005) (11 pages) doi:10.1115/1.2175161 History: Received February 14, 2005; Revised October 02, 2005
Article
References
Figures
Citing Articles

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=u∕ϕ), and new parameters Rϕ=Rτ∕ϕ termed as the roughness friction Reynolds number, Reϕ=Re∕ϕ 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+=Zuτ∕ν and u+=u∕uτ). 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 h∕δ (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.
FIGURES IN THIS ARTICLE
<>
Copyright © 2006 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

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., 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., 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  [CrossRef], 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.  [CrossRef], 151 , pp. 171–183.
7
George, W., and Castillo, L., 1997, “Zero Pressure Gradient Turbulent Boundary Layer,” Appl. Mech. Rev., 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.  [CrossRef], 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  [CrossRef], 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., 41 (4), pp. 565–572.
15
Buschmann, M. H., and Gad-el-Hak, K., 2003, “Generalized Logarithmic Law and its Consequences,” AIAA J., 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, 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.  [CrossRef], 421 , pp. 115–145.
19
Schultz, M. P., and Myers, A., 2003, “Comparison of Three Roughness Function Determination Methods,” Exp. Fluids  [CrossRef], 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., 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.  [CrossRef], 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  [CrossRef], 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., 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 FlowCan. J. Civ. Eng., 29 , pp. 256–266.
29
Clauser, F. H., 1954, “Turbulent Boundary Layers in Adverse Pressure Gradients,” J. Aeronaut. Sci., 4 , 91–108.
30
Hama, F. R., 1954, “Boundary Layer Characteristics for Smooth and Rough Surfaces,” Soc. Nav. Archit. Mar. Eng., Trans., 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  [CrossRef], 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  [CrossRef], 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), 11 , pp. 133–156.
37
Moody, L. F., 1944, “Friction Factor for Pipe Flow,” Trans. ASME, 66 , pp. 676–684.
38
Zagarola, M. V., and Smits, A. J., 1998, “Mean Flow Scaling in Turbulent Pipe Flow,” J. Fluid Mech.  [CrossRef], 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  [CrossRef], 15 , pp. 3079–3089.
41
Tennekes, H., 1968, “Outline of a Second Order Theory of Turbulent Pipe Flow,” AIAA J., 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., 61 , pp. 23–31.
43
Afzal, N., 1976, “Millikan’s Argument at Moderately Large Reynolds Numbers,” Phys. Fluids  [CrossRef], 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.  [CrossRef], 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., 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.

Figures

Grahic Jump Location
+Comparison of power law constant A relation 27 against inverse of the power law index α with the experimental data of fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation——A=0.92∕α+2.2.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of power law constant A relation 27 against inverse of the power law index α with the experimental data of fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation——A=0.92∕α+2.2.
Figure 4

Comparison of power law constant A relation 27 against inverse of the power law index α with the experimental data of fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation——A=0.92∕α+2.2.

Grahic Jump Location
+Comparison of the power law index α relation 40 against roughness friction Reynolds number Rϕ with the experimental data of the fully developed turbulent flow in transitional rough pipe of Nikuradse for various values of δ∕h in pipes. Proposed relation—– α=1∕lnRϕ.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of the power law index α relation 40 against roughness friction Reynolds number Rϕ with the experimental data of the fully developed turbulent flow in transitional rough pipe of Nikuradse for various values of δ∕h in pipes. Proposed relation—– α=1∕lnRϕ.
Figure 5

Comparison of the power law index α relation 40 against roughness friction Reynolds number Rϕ with the experimental data of the fully developed turbulent flow in transitional rough pipe of Nikuradse for various values of δ∕h in pipes. Proposed relation—– α=1∕lnRϕ.

Grahic Jump Location
+The power law velocity distribution in log-log plots for sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=60: (a) Traditional inner power law velocity profile u+=CZ+α, in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against traditional smooth wall variable Z+ according to the power law relation u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall variable ζ for velocity profile u+, based on proposed inner power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα, in outer variables (u+,Y).
View Large  |  Save Figure  |  Download Slide (.ppt)
The power law velocity distribution in log-log plots for sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=60: (a) Traditional inner power law velocity profile u+=CZ+α, in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against traditional smooth wall variable Z+ according to the power law relation u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall variable ζ for velocity profile u+, based on proposed inner power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα, in outer variables (u+,Y).
Figure 1

The power law velocity distribution in log-log plots for sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=60: (a) Traditional inner power law velocity profile u+=CZ+α, in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against traditional smooth wall variable Z+ according to the power law relation u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall variable ζ for velocity profile u+, based on proposed inner power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα, in outer variables (u+,Y).

Grahic Jump Location
+The power law velocity distribution in log-log plots for sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=15: (a) Traditional inner power law velocity profile u+=CZ+α, in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against traditional smooth wall variable Z+ according to the power law relation u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall variable ζ for velocity profile u+, based on proposed inner power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα, in outer variables (u+,Y).
View Large  |  Save Figure  |  Download Slide (.ppt)
The power law velocity distribution in log-log plots for sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=15: (a) Traditional inner power law velocity profile u+=CZ+α, in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against traditional smooth wall variable Z+ according to the power law relation u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall variable ζ for velocity profile u+, based on proposed inner power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα, in outer variables (u+,Y).
Figure 2

The power law velocity distribution in log-log plots for sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=15: (a) Traditional inner power law velocity profile u+=CZ+α, in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against traditional smooth wall variable Z+ according to the power law relation u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall variable ζ for velocity profile u+, based on proposed inner power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα, in outer variables (u+,Y).

Grahic Jump Location
+The power law velocity distribution u∕Uc=Yα from relations 52,52 in the sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=60
View Large  |  Save Figure  |  Download Slide (.ppt)
The power law velocity distribution u∕Uc=Yα from relations 52,52 in the sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=60
Figure 3

The power law velocity distribution u∕Uc=Yα from relations 52,52 in the sand-grain rough pipe data of Nikuradse for various values of h+ with δ∕h=60

Grahic Jump Location
+Comparison of the power law constant A relation 40 against roughness friction Reynolds number Rϕ with the experimental data of the transitional rough pipe data of Nikuradse for various values of δ∕h. values in pipes) for Nikuradse data. Proposed relation——A=0.9lnRϕ+2.6.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of the power law constant A relation 40 against roughness friction Reynolds number Rϕ with the experimental data of the transitional rough pipe data of Nikuradse for various values of δ∕h. values in pipes) for Nikuradse data. Proposed relation——A=0.9lnRϕ+2.6.
Figure 6

Comparison of the power law constant A relation 40 against roughness friction Reynolds number Rϕ with the experimental data of the transitional rough pipe data of Nikuradse for various values of δ∕h. values in pipes) for Nikuradse data. Proposed relation——A=0.9lnRϕ+2.6.

Grahic Jump Location
+Comparison of the power law index α against friction factor λ based on relation 37 with fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation——α=0.88λ.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of the power law index α against friction factor λ based on relation 37 with fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation——α=0.88λ.
Figure 7

Comparison of the power law index α against friction factor λ based on relation 37 with fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation——α=0.88λ.

Grahic Jump Location
+Comparison of power law constant A against parameter λ−1∕2 based on inverse nondimensional friction factor from relation 37 with for fully developed turbulent flow in transitional rough pipe data of Nikuradse for varios values of δ∕h in pipes. Proposed relation——A=0.92∕λ+2.6.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of power law constant A against parameter λ−1∕2 based on inverse nondimensional friction factor from relation 37 with for fully developed turbulent flow in transitional rough pipe data of Nikuradse for varios values of δ∕h in pipes. Proposed relation——A=0.92∕λ+2.6.
Figure 8

Comparison of power law constant A against parameter λ−1∕2 based on inverse nondimensional friction factor from relation 37 with for fully developed turbulent flow in transitional rough pipe data of Nikuradse for varios values of δ∕h in pipes. Proposed relation——A=0.92∕λ+2.6.

Grahic Jump Location
+Comparision of λe∕λ, the ratio of the experimental to predicted values of friction factors against the roughness friction Reynolds number Rϕ with fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation—– λe∕λ=1.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparision of λe∕λ, the ratio of the experimental to predicted values of friction factors against the roughness friction Reynolds number Rϕ with fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation—– λe∕λ=1.
Figure 9

Comparision of λe∕λ, the ratio of the experimental to predicted values of friction factors against the roughness friction Reynolds number Rϕ with fully developed turbulent flow in transitional rough pipe data of Nikuradse for various values of δ∕h in pipes. Proposed relation—– λe∕λ=1.

Grahic Jump Location
+Comparison of the friction factor λ against the roughness friction Reynolds number Rϕ=Rr∕ϕ from prediction 55 with data of Nikuradse for various values of δ∕h for fully developed turbulent flow in transitional rough pipes. Proposed relation—–1∕λ=2log10(Rϕ32)−0.8.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of the friction factor λ against the roughness friction Reynolds number Rϕ=Rr∕ϕ from prediction 55 with data of Nikuradse for various values of δ∕h for fully developed turbulent flow in transitional rough pipes. Proposed relation—–1∕λ=2log10(Rϕ32)−0.8.
Figure 10

Comparison of the friction factor λ against the roughness friction Reynolds number Rϕ=Rr∕ϕ from prediction 55 with data of Nikuradse for various values of δ∕h for fully developed turbulent flow in transitional rough pipes. Proposed relation—–1∕λ=2log10(Rϕ32)−0.8.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Other Components and Variations
Axial-Flow Compressors> Chapter 13
Incremental Model Adjustment
Nonlinear Regression Modeling for Engineering Applications: Modeling, Model Validation, and Enabling Design of Experiments
The Design and Implement of Remote Inclinometer for Power Towers Based on MXA2500G/GSM
International Conference on Mechanical and Electrical Technology, 3rd, (ICMET-China 2011), Volumes 1–3
Natural Gas Transmission
Pipeline Design & Construction: A Practical Approach, Third Edition> Chapter 3
Introduction
Development of Reliability-Based Load and Resistance Factor Design (LRFD) Methods for Piping> Chapter 1
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
This site uses cookies. By continuing to use our website, you are agreeing to our privacy policy. | Accept
Sign In