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Research Papers: Fundamental Issues and Canonical Flows

# Alternate Scales for Turbulent Boundary Layer on Transitional Rough Walls: Universal Log Laws

[+] Author and Article Information
Noor Afzal

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

J. Fluids Eng 130(4), 041202 (Apr 01, 2008) (16 pages) doi:10.1115/1.2844583 History: Received March 30, 2007; Revised November 02, 2007; Published April 01, 2008

## Abstract

The present work deals with four new alternate transitional surface roughness scales for description of the turbulent boundary layer. The nondimensional roughness scale $ϕ$ is associated with the transitional roughness wall inner variable $ζ=Z+∕ϕ$, the roughness friction Reynolds number $Rϕ=Rτ∕ϕ$, and the roughness Reynolds number $Reϕ=Re∕ϕ$. The two layer theory for turbulent boundary layers in the variables, mentioned above, is presented by method of matched asymptotic expansions for large Reynolds numbers. The matching in the overlap region is carried out by the Izakson–Millikan–Kolmogorov hypothesis, which gives the velocity profiles and skin friction universal log laws, explicitly independent of surface roughness, having the same constants as the smooth wall case. In these alternate variables, just above the wall roughness level, the mean velocity and Reynolds stresses are universal and do not depend on surface roughness. The extensive experimental data provide very good support to our universal relations. There is no universality of scalings in traditional variables and different expressions are needed for inflectional type roughness, monotonic Colebrook–Moody roughness, $k$-type roughness, $d$-type roughness, etc. In traditional variables, the velocity profile and skin friction predictions for the inflectional roughness, $k$-type roughness, and $d$-type roughness are supported well by the extensive experimental data. The pressure gradient effect from the matching conditions in the overlap region leads to the universal composite laws, which for weaker pressure gradients yields log laws and for strong adverse pressure gradients provides the half-power laws for universal velocity profiles and in traditional variables the additive terms in the two situations depend on the wall roughness.

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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures

Figure 1

The velocity distribution on k-type roughness from data of Kameda (32) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 2

The log law velocity distribution on d-type roughness from data of Osaka and Mochizuki (31) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 3

The log law velocity distribution due to rod roughness data of Smalley (34) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 4

The log law velocity distribution on 60- and 220-grid sand grain data of Schultz and Flack (36) and epoxy roughness data of Schultz and Myers (35) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 5

The log law velocity distribution on uniform spheres and uniform spheres with grit data of Schultz and Flack (22) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 6

The log law velocity distribution on machine honed surface roughness data of Schultz and Flack (19) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 7

The log law velocity distribution on river bed roughness from data of Rahman and Webster (37) in semilog plots: (a) Inner smooth wall variables (u+,Z+). (b) Velocity profile shifted by the roughness function u++ΔU+ against smooth wall variable Z+. (c) Our universal variables for inner transitional rough wall (u+,ζ). (d) Alternate universal inner rough wall variables (u+,Z∕Z0). (e) Outer velocity defect variables ((Uc−u)∕uτ,Y).

Figure 8

Comparison of the roughness scales data with present predictions. (a) The roughness function ΔU+ against wall roughness parameter h+ for various values of δ∕h. (b) Roughness scale ϕ against wall roughness parameter h+ for various values of δ∕h. (c) Roughness scale Z0∕h against wall roughness parameter h+ for various values of δ∕h. The lines marked S is the inflectional roughness (j=11) and marked C is for Colebrook monotonic roughness (j=0) from predictions Eq. 40 for roughness scale ϕ, Eq. 41 for roughness function ΔU+, and Eq. 42 for roughness length Z0+. Line marked KD is the prediction 51 for k-type and d-type roughnesses.

Figure 9

The log law intercept BT and Bt for boundary layer data for various values of h+ and δ∕h. Prediction lines line marked S are for sand grain roughness and marked C is for Colebrook commercial roughness. Line marked KD is the prediction for k-type and d-type roughnesses.

Figure 10

Comparison of universal skin friction Cf predictions and experimental data for turbulent flow boundary layer on transitional rough walls. (a) Skin friction coefficient Cf against roughness Reynolds number Rϕ=Rτ∕ϕ data and universal prediction 53, 1∕Cf=1.76lnRϕ+5.09. (b) Skin friction coefficient Cf against roughness Reynolds number Reϕ=Re∕ϕ data and universal prediction 54, 1∕Cf=1.76ln(ReϕCf)+4.48.

Figure 11

Comparison of the skin friction Cf versus traditional Reynolds number Re prediction 55 for δ∕h=15, 30, 60, 120, 200, and 400 with experimental data for inflectional wall roughness

Figure 12

Comparison of the skin friction Cf versus traditional Reynolds number Re with prediction 52 for δ∕h=20, 60, 100, 125, 200, and 400 with experimental data for k-type and d-type wall roughnesses

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