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Research Papers: Fundamental Issues and Canonical Flows

Analysis of a Power Law and Log Law for a Turbulent Wall Jet Over a Transitional Rough Surface: Universal Relations

[+] Author and Article Information
Noor Afzal

Faculty of Engineering,  Aligarh Muslim University, Aligarh - 202002, India

Abu Seena

Department of Mechanical Engineering,  Korea Advanced Institute of Science and Technology 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, South Korea

J. Fluids Eng 133(9), 091201 (Sep 08, 2011) (14 pages) doi:10.1115/1.4004763 History: Received January 06, 2011; Revised July 25, 2011; Published September 08, 2011; Online September 08, 2011

The power law and log law velocity profiles and an integral analysis in a turbulent wall jet over a transitional rough surface have been proposed. Based on open mean momentum Reynolds equations, a two layer theory for large Reynolds numbers is presented and the matching in the overlap region is carried out by the Izakson-Millikan-Kolmogorov hypothesis. The velocity profiles and skin friction are shown to be governed by universal log laws as well as by universal power laws, explicitly independent of surface roughness, having the same constants as a fully smooth surface wall jet (or fully rough surface wall jet, as appropriate). The novel scalings for stream-wise variations of the flow over a rough wall jet have been analyzed, and best fit relations for maximum wall jet velocity, boundary layer thickness at maxima of wall jet velocity, the jet half width, the friction factor, and momentum integral are supported by the experimental data. There is no universality of scalings in traditional variables, and different expressions are needed for transitional roughness. The experimental data provides very good support to our universal relations proposed in terms of alternate variables.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Transverse length scales associated with a wall jet on a transitional rough surface without a free stream

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Figure 2

The outer layer power law velocity profile (u/Um vs z/δm) in a turbulent wall jet over a transitional rough surface from data of Smith [4]

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Figure 3

The turbulent wall jet over a transitional rough surface for the log law velocity profile from data of Smith [4] in inner wall layer variables: (a) Proposed universal variables (u+,ζ = z+/φ) and the university law of the wall u+=k-1lnζ+B. (b) Alternate universal variables (u++ΔU+,z+) and universal law of the wall u++ΔU+=k-1lnz++B. (c) Traditional smooth wall variables (u+,z+) and law of the wall u+=k-1lnz+B-ΔU+, which depends on ΔU+ roughness function.

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Figure 4

The turbulent wall jet on a rough flat surface from data of Smith [4] in outer wall layer variables ((u-Um)/uτ,Y=z/δ) and the domain of the outer log law (u-Um)/uτ=k-1lnY-D in (a) linear variables and (b) semi-log variables

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Figure 5

The turbulent wall jet on a rough flat surface from data of Smith [4] in outer wall layer variables ((u-Um)/uτ,Ym=z/δm) and the domain of the outer log law (u-Um)/uτ=k-1lnYm-Dm in (a) linear variables and (b) semi-log variables

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Figure 6

(a) The roughness scale φ vs ke+ and (b) the roughness function ΔU+ vs ke+ from data of Smith [4] and Tachie [5] for the turbulent wall jet on a rough flat surface and the proposed monotonic relations

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Figure 7

The turbulent wall jet over a transitional rough surface power law velocity profile from data of Smith [4] in inner wall layer variables: (a) Proposed universal variables (u+,ζ=z+/δ) and the universal power law of the wall u+=Aζα. (b) Alternate universal variables (u++ΔU+,z+) and universal law of the wall u++ΔU+=Cz+α. (c) Traditional smooth wall variables (u+,z+) and law of the wall u+=Cz+α-ΔU+, which depends on ΔU+ roughness function.

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Figure 12

The stream-wise variation of the length scales from the data of Smith [4] for the turbulent wall jet on a rough flat surface. (a) Boundary layer thickness δ based on the location of Um/2, the half of the maximum wall jet velocity, and the best fit relation δb/ke2=0.089(xb/ke2)0.997. (b) Boundary layer thickness based on the location of the maximum wall jet velocity, the best fit relation δmb/ke2=0.0108(xb/ke2)0.979, and the correlation proposed by Hogg [2].

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Figure 13

The stream-wise variation of the friction factor from the data of Smith [4] for the turbulent wall jet on a rough surface. Best fit relation Um/uτ=8.86(xb/ke2)0.054.

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Figure 14

The stream-wise variation of the momentum M from the data of Smith [4] for the turbulent wall jet on a rough surface. The best fit relation (J-M)/J=0.147(xb/ke2)-0.09.

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Figure 11

The stream-wise variation of Um, the maximum wall jet velocity from the data of Smith [4] for the turbulent wall jet on a rough surface, the best fit relation Umke/U0b=3.81(xb/ke2)-0.49, and the correlation proposed by Hogg [2]

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Figure 10

The coefficient of skin friction Cf versus (a) Rmτ/φ and (b) Rm/φ from transitional rough wall jet data

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Figure 9

Comparison of the lowest order outer wake layer velocity profile prediction u/Um=1-tanh2(ΩY), Ω=0.881 based on the outer layer eddy viscosity closure model with wall jet data of Smith [4] on a rough surface

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Figure 8

Comparison of power law constants from the experimental data of turbulent wall jet on a transitional rough wall: (a) Power law prefactor A against the inverse of the power law index α. Proposed relation A=0.92/α+2.1. (b) Power law index α against the roughness friction Reynolds number Rφ. Proposed relation α=1/lnRφ. (c) The power law prefactor A against the roughness friction Reynolds number Rφ. Proposed relation A=0.92lnRφ+1.6. (d) Comparison of the power law index α against the skin friction coeffecient ε for turbulent wall jet on transitional rough walls. Proposed relation α=2.5ε. (e) Comparison of the power law constant A against parameter exp(-1)/ε based on the inverse ε with turbulent wall jet on transitional rough walls. Proposed relation A=exp(-1)/ε-0.7.

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