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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
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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.
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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).
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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).
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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
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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 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.
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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.
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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
+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.
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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.
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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.
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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.
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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.
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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.

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