0
TECHNICAL PAPERS

Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow

[+] Author and Article Information
Noor Afzal

Department of Mechanical Engineering, Aligarh University, Aligarh 202002, Indianoor.afzal@yahoo.com

J. Fluids Eng 129(10), 1255-1267 (May 16, 2007) (13 pages) doi:10.1115/1.2776961 History: Received June 25, 2006; Revised May 16, 2007

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2007 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Comparison of the roughness characteristics of the pipes from P2-P7 data of Sletfjerding, a∕h=7190 data of Shockling, and a∕h=507 data of Nikuradse for inflectional roughness with present prediction 29 for j=11 shown by a solid line. Comparison of P1 data of Sletfjerding with Colebrook monotonic roughness prediction 29 with j=0 shown by a dash line. (a) Roughness function ΔU+ against h+. (b) Roughness scale ϕ against h+.

Grahic Jump Location
Figure 2

Comparison of the roughness characteristics of the pipes from P2-P7 data of Sletfjerding, a∕h=7190 data of Shockling, and a∕h=507 data of Nikuradse showing inflectional roughness and P1 data of Sletfjerding showing monotonic roughness with present work. (a) Friction factor λ with roughness Reynolds number Re∕ϕ from the data showing universal behavior, explicitly independent of transitional pipe roughness. (—) Present universal relation 25. (b) Friction factor λ with traditional Reynolds number Re from data showing nonuniversal behavior that depends on transitional pipe roughness. (—) Present inflectional roughness prediction 34 with j=11, (----) Colebrook monotonic roughness prediction 34 with j=0.

Grahic Jump Location
Figure 3

Comparison of the roughness characteristics of the pipes from P2-P7 data of Sletfjerding, a∕h=7190 data of Shockling, and a∕h=507 data of Nikuradse showing inflectional roughness and P1 data of Sletfjerding showing monotonic roughness with present work. (a) Friction factor λ with roughness Reynolds number Re∕ϕ from data showing universal behavior, explicitly independent of transitional pipe roughness. (—) Present universal relation 25. (b) Friction factor λ with traditional Reynolds number Re from data showing nonuniversal behavior that depends on transitional pipe roughness. (—) Inflectional roughness, present prediction 34 with j=11 and comparison with the data of Sletfjerding (P2-P8), Shockling and Nikuradse, (----) Colebrook monotonic roughness prediction 34 with j=0 and comparison with the data of Sletfjerding (P1).

Grahic Jump Location
Figure 4

Comparison of the roughness characteristics of the pipes from P2-P8 data of Sletfjerding, a∕h=7190 data of Shockling, and a∕h=507 data of Nikuradse showing inflectional roughness and P1 data of Sletfjerding showing monotonic roughness with present work. (a) Friction factor 1∕λ with roughness Reynolds number Re∕ϕ from data showing universal behavior, explicitly independent of transitional pipe roughness. (—) Present universal relation 25. (b) Friction factor λ with traditional Reynolds number Re from data showing nonuniversal behavior that depends on transitional pipe roughness. (—) Inflectional roughness, present prediction 34 with j=11 and comparison with the data of Sletfjerding (P2-P8), Shockling and Nikuradse, (----) Colebrook monotonic roughness prediction 34 with j=0 and comparison with the data of Sletfjerding (P1).

Grahic Jump Location
Figure 5

Comparison of the present prediction 31 for bT=λ−1∕2−2log(a∕h) for inflectional roughness (j=11) and Colebrook monotonic roughness (j=0) with data of Sletfjerding, Shockling, and Nikuradse.

Grahic Jump Location
Figure 6

Our prediction of the relationship between alternate roughnesses: Ra the arithmetic mean roughness, RZ the mean peak to valley height roughness, Rq the root mean square (rms) roughness, and Rq∕H the height-texture roughness, with respect to h the equivalent sand grain roughness from data of Sletfjerding and Gudmundsson.

Grahic Jump Location
Figure 7

Second order effect of order (Reϕλ)−1: The friction factor departure function Λ≡1∕λ−Alog(Reλ) against (Reϕλ)−1 from Sletfjerding’s (3) inflectional roughness data P2-P8 a∕h=402–3280 (▴), Shockling’s (28) machine-honed surface roughness data a∕h=7190 (●), Nikuradse’s (1) sand grain roughness data a∕h=15–507 (∎), and (⋯ ⋯) Blasius-type 1∕4 power law for transitional roughness relation 24, along with fully smooth pipe data (McKeon (10) (엯), Oregon data (38) (◻), and Patel and Head (33) (▵)). (a) Computations from the data based on Prandtl constant A=2. (—) Present prediction Λ=−0.88+75(Reϕλ)−1, (---) Prandtl-type relation 25 with Λ=−0.8. (b) Computations from the data based on McKeon constant A=1.93. (—) Present prediction Λ=−0.537−85(Reϕλ)−1, (---) McKeon-type relation 26 with Λ=−0.537.

Grahic Jump Location
Figure 8

Second order mesolayer effect of order (Reϕλ)−1∕2: The friction factor departure function Λ≡1∕λ−Alog(Reλ) against (Reϕλ)−1∕2 from Sletfjerding’s (3) inflectional roughness data P2-P8 a∕h=402–3280 (▴), Shockling’s (28) machine-honed surface roughness data a∕h=7190 (●), Nikuradse’s (1) sand grain roughness data a∕h=15–507 (∎), and (⋯ ⋯) Blasius-type 1∕4 power law for transitional roughness relation 24 along with fully smooth pipe data (McKeon (10) (엯), Oregon data (38) (◻), and Patel and Head (33) (▵)). (a) Computations from the data based on Prandtl constant A=2. (—) Present prediction Λ=−0.88+2.2(Reϕλ)−1∕2, (---) Prandtl-type relation 25 with Λ=−0.8. (b) Computations from the data based on McKeon constant A=1.93. (—) Present prediction Λ=−0.537−3.8(Reϕλ)−1∕2, (---) McKeon-type relation 26 with Λ=−0.537.

Grahic Jump Location
Figure 9

Second order effect with substantive mesolayer: The friction factor departure function Λ≡1∕λ−Alog(Reλ) with mesolayer A=(Ai+A0)∕(32log10e)≈3 from Sletfjerding’s (3) inflectional roughness data P2-P8 a∕h=402–3280 (▴), Shockling’s (28) machine-honed surface roughness data a∕h=7190 (●), Nikuradse’s (1) sand grain roughness data a∕h=15–507 (∎), and (⋯⋯) Blasius-type 1∕4 power law for transitional roughness relation 24 along with fully smooth pipe data (McKeon (10) (엯), Oregon data (38)(◻), and Patel and Head (33) (▵)). (a) Λ versus (Reϕλ)−1 and (—) Present prediction Λ=−4.2+325(Reϕλ)−1. (b) Λ versus (Reϕλ)−1∕2 and (—) present prediction Λ=−5+33.4(Reϕλ)−1∕2.

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
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
Sign In