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TECHNICAL PAPERS

# Power Law Velocity Profile in the Turbulent Boundary Layer on Transitional Rough Surfaces

[+] Author and Article Information
Noor Afzal

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

J. Fluids Eng 129(8), 1083-1100 (Mar 04, 2007) (18 pages) doi:10.1115/1.2746902 History: Received January 29, 2006; Revised March 04, 2007

## Abstract

A new approach to scaling of transitional wall roughness in turbulent flow is introduced by a new nondimensional roughness scale $ϕ$. This scale gives rise to an inner viscous length scale $ϕν∕uτ$, inner wall transitional variable, roughness friction Reynolds number, and roughness Reynolds number. The velocity distribution, just above the roughness level, turns out to be a universal relationship for all kinds of roughness (transitional, fully smooth, and fully rough surfaces), but depends implicitly on roughness scale. The open turbulent boundary layer equations, without any closure model, have been analyzed in the inner wall and outer wake layers, and matching by the Izakson-Millikan-Kolmogorov hypothesis leads to an open functional equation. An alternate open functional equation is obtained from the ratio of two successive derivatives of the basic functional equation of Izakson and Millikan, which admits two functional solutions: the power law velocity profile and the log law velocity profile. The envelope of the skin friction power law gives the log law, as well as the power law index and prefactor as the functions of roughness friction Reynolds number or skin friction coefficient as appropriate. All the results for power law and log law velocity and skin friction distributions, as well as power law constants are explicitly independent of the transitional wall roughness. The universality of these relations is supported very well by extensive experimental data from transitional rough walls for various different types of roughnesses. On the other hand, there are no universal scalings in traditional variables, and different expressions are needed for various types of roughness, such as inflectional roughness, monotonic roughness, and others. To the lowest order, the outer layer flow is governed by the nonlinear turbulent wake equations that match with the power law theory as well as log law theory, in the overlap region. These outer equations are in equilibrium for constant value of $m$, the pressure gradient parameter, and under constant eddy viscosity closure model, the analytical and numerical solutions are presented.

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## Figures

Figure 1

The power law velocity distribution in log–log plots from data of Kameda, Osaka, and Mochizuki (45) for k-type roughness in the turbulent boundary layer: (a) Traditional inner power law u+=CZ+α. (b) Velocity profile shifted by the roughness function inner power law u++ΔU+=CZ+α. (c) Proposed inner transitionally rough wall power law velocity profile u+=Aζα. (d) Proposed outer power law velocity profile u+=A1Yα.

Figure 2

The power law velocity distribution in log–log plots from data of Osaka and Mochizuki (46) for d-type roughness in the turbulent boundary layer. Legend same as in Figs.  1111.

Figure 3

The power law velocity distribution in log–log plots from data of Smalley (48) for rod roughness in the turbulent boundary layer. Legend same as in Figs.  1111.

Figure 4

The power law velocity distribution in log–log plots for turbulent boundary layer data of Schultz and Flack (49) for 60-grit and 120-grid sand grain roughness, and Schultz and Myers (50) for epoxy roughness. Legend same as in Figs.  1111.

Figure 5

The power law velocity distribution in log–log plots for turbulent boundary layer from data of Schultz and Flack (51) data for uniform spheres and uniform spheres with grit roughness. Legend same as in Figs.  1111.

Figure 6

The power law velocity distribution in log–log plots from data of Rahman and Webster (52) for channel bed roughness. Legend same as in Figs.  1111.

Figure 7

Comparison of power law constant A from Eq. 108 against inverse of the power law index α with the experimental data of turbulent boundary layer on transitional rough wall. Proposed relation A=0.92∕α+2.1.

Figure 8

Comparison of the power law index α from Eq. 107 against roughness friction Reynolds number Rϕ with the experimental data of the turbulent boundary layer on transitional rough walls. Proposed relation: α=1∕lnRϕ.

Figure 9

Comparison of the power law constant A from Eq. 109 against roughness friction Reynolds number Rϕ with the experimental data of turbulent boundary layer on transitional rough walls. Proposed relation: A=0.92lnRϕ+1.6.

Figure 10

Comparison of the power law index α against skin fiction coefficient Cf based on Eq. 110 for turbulent boundary layer on transitional rough walls. Proposed relation: α=2.5ϵ∕(1−6.2ϵ).

Figure 11

Comparison of power law constant A relation against parameter exp(−1)∕ϵ based on inverse nondimensional friction factor from Eq. 111 with turbulent boundary layer on transitional rough walls. Proposed relation: A=exp(−1)∕ϵ−0.7.

Figure 12

Comparison of skin friction coefficient Cf against the roughness friction Reynolds number Rϕ=Rτ∕ϕ from Eq. 114 for turbulent flow boundary layer on transitional rough wall. Proposed relation: 1∕Cf=1.76lnRϕ+5.09.

Figure 13

Comparison of the roughness scales data with present predictions. (a) Roughness scale ϕ against wall roughness parameter h+ for various values of δ∕h. (b) The roughness function ΔU+ against wall roughness parameter h+ for various values of δ∕h. The line marked “S” is the inflectional roughness (j=11) and marked “C” is for Colebrook monotonic roughness (j=0) from prediction equation 103 for roughness scale ϕ and roughness function ΔU+. Line marked “KD” is the prediction Eq. 104 for k-type and d-type roughness.

Figure 14

Comparison of velocity profile from solution of lowest order outer nonlinear wake layer Falkner-Skan equations 9,10,11,12 based on Clauser constant eddy viscosity model, for m=−0.165858(β=−0.198838) for skin friction Cf=0, and slip velocity bS=0 with the wake function of Coles (60)

Figure 15

The wake- and jet-like solutions of the velocity distribution from lowest order outer nonlinear wake layer Falkner-Skan equations 9,10,11,12 for Cf=0, based on Clauser constant eddy viscosity model

Figure 16

The skin friction parameter Cf∕αc and wall slip velocity parameter bS=US∕Ue from solution outer nonlinear wake layer Falkner-Skan equations 9,10,11,12, based on Clauser constant eddy viscosity model

Figure 17

Comparison of the perturbed wake function W1(Y) for outer shallow wake layer proposed by Afzal (38) with area I1=∫01W1(Y)dY=0 with Lewkowicz (42) for I1=0.0166 and Finley (43) and Granville (44) for I1=0.0833

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