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TECHNICAL PAPERS

Alternate Scales for Turbulent Flow in Transitional Rough Pipes: Universal Log Laws

[+] Author and Article Information
Noor Afzal

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

Abu Seena

Department of Mechanical Engineering,  Aligarh University, Aligarh 202002, India

J. Fluids Eng 129(1), 80-90 (Jun 22, 2006) (11 pages) doi:10.1115/1.2375129 History: Received February 06, 2006; Revised June 22, 2006

Abstract

In transitional rough pipes, the present work deals with alternate four new scales, the inner wall transitional roughness variable $ζ=Z+∕ϕ$, associated with a particular roughness level, defined by roughness scale $ϕ$ connected with roughness function $▵U+$, the roughness friction Reynolds number $Rϕ$ (based on roughness friction velocity), and roughness Reynolds number $Reϕ$ (based on roughness average velocity) where the mean turbulent flow, little above the roughness sublayer, does not depend on pipes transitional roughness. In these alternate variables, a two layer mean momentum theory is analyzed by the method of matched asymptotic expansions for large Reynolds numbers. The matching of the velocity profile and friction factor by Izakson-Millikan-Kolmogorov hypothesis gives universal log laws that are explicitly independent of pipe roughness. The data of the velocity profile and friction factor on transitional rough pipes provide strong support to universal log laws, having the same constants as for smooth walls. There is no universality of scalings in traditional variables and different expressions are needed for various types of roughness, as suggested, for example, with inflectional-type roughness, monotonic Colebrook-Moody roughness, etc. In traditional variables, the roughness scale, velocity profile, and friction factor prediction for inflectional pipes roughness are supported very well by experimental data.

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Figures

Figure 1

The log law velocity profile for transitional roughness δ∕h=60 for various values of h+ in the range 31≤h+≤1230.3 from sand grain data of Nikuradse (1933): (a) Traditional inner wall law in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++▵U+ with smooth wall variable Z+. (c) Proposed inner transitional rough wall in universal variables (u+,ζ). (d) Outer velocity defect variables (Uc+−u+,Y).

Figure 2

The log law velocity profile for transitional roughness δ∕h=60 for various values of h+ in the range 8≤h+≤369.8 from sand grain data of Nikuradse (1933): (a) Traditional inner wall law in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++▵U+ with smooth wall variable Z+. (c) Proposed inner transitional rough wall in universal variables (u+,ζ). (d) Outer velocity defect variables (Uc+−u+,Y).

Figure 3

The log law velocity profile for transitional roughness δ∕h=7190 for various values of h+ in the range 0.1≤h+≤44.46 from machine surface roughness super pipe data of Shockling (2005): (a) Traditional inner wall law in smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++▵U+ with smooth wall variable Z+. (c) Proposed inner transitional rough wall in universal variables (u+,ζ). (d) Outer velocity defect variables (Uc+−u+,Y).

Figure 4

The transitional roughness characteristics from Nikuradse data of sand grain pipes roughness and Shockling data of machined honed surface roughness in Princeton’s superpipe: (a) Roughness function ▵U+ against h+. (b) Roughness scale ϕ against h+. Prediction lines 41 and 39 marked S is for inflectional roughness and marked C is for Colebrook commercial roughness.

Figure 5

The log law intercepts BT and Bt from Nikuradse data of sand grain roughness and Shockling data of machined honed surface roughness for various values of h+ and δ∕h. Sh-R denotes the predictions of BT and Bt from Shocking roughness function and roughness (▵U+, h+) data and constants k=0.4 and B=5.5. Prediction lines 43,43 marked S is for inflectional roughness and marked C is for Colebrook commercial roughness.

Figure 6

Comparison of the universal friction factor λ of transitional rough pipes data of Nikuradse and Shockling and present predictions: (a) Friction factor λ vs roughness friction Reynolds number Rϕ=Rτ∕ϕ data and relation 34——1∕λ=2log10(Rϕ32)−0.8. (b) Friction factor λ vs roughness Reynolds number Reϕ=Re∕ϕ data and relation 33——1∕λ=2log10(Reϕλ)−0.8.

Figure 7

Comparison of the friction factor λ of transitional rough pipes data of Nikuradse and Shockling and present predictions: (a) Friction factor λ vs friction Reynolds number Rτ data and relation (69). (b) Friction factor λ vs Reynolds number Re data and relation (70).

Figure 8

Comparison of proposed predictions of the friction factor λ with experimental data. (a) Fiction factor λ with roughness Reynolds number function Reϕλ showing the universal relationship. (b) Fiction factor λ with traditional Reynolds number function Reλ. Smooth pipes data: McKeon (2003), Oregon data (McKeon 2004), Patel and Head (1968), Blasius (1912) relation. Transitional pipe roughness data: Shockling (2005) for machine honed surface roughness and Nikuradse (1933) sand grain roughness data.

Figure 9

Comparison of maxima in Reynolds shear stress prediction of Afzal (1982) with the experimental data of Zanoun (2003) for smooth pipe and smooth channel and DNS data of Abe (2004) for channel. (a) The prediction 55 for location of the maxima shear stress. (b) The prediction 55 for magnitude of maximum shear stress.

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