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Hydraulic characterization of sandpaper roughness

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Abstract

Experiments on 220, 320 and 500-grit sandpaper surfaces were conducted in a high Reynolds number turbulent channel flow facility with an 8:1 aspect ratio. Pressure drop measurements in the fully developed region yielded the skin-friction curves for Reynolds number based on the bulk mean velocity (Rem) up to 2.5 × 105. Detailed velocity profiles were also obtained for the 320-grit sandpaper at friction Reynolds number (Reτ) from 1000 to 6000 using laser Doppler velocimetry (LDV). The skin-friction curves and roughness functions for the sandpaper surfaces show many similarities to and some differences from uniform sand roughness. The most notable difference is that the equivalent sand roughness height (ks) for the sandpaper is several times larger than its median grit size. Possible reasons for this disparity are discussed. Mean flow and turbulence results on 320-grit sandpaper indicate similarity between the rough and smooth walls in the outer layer (y/h > 0.15). ks for each sandpaper studied is compared to predicted results from existing empirical correlations based on statistics of the surface texture. A correlation for ks based on the rms roughness height (krms) and the skewness of the surface elevation probability density function (Sk) shows the most promise.

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Data availability

The data in this paper will be available at http://roughnessdatabase.org.

Abbreviations

A :

Log-law intercept for a smooth wall = 4.27

B s :

Log-law constant for a rough wall

B s(∞):

Log-law constant for a fully rough wall = 8.5

C f :

Skin-friction coefficient \(= \tau_{w} {/}(1{/}2\rho \overline{U}^{2} )\)

d :

LDV probe volume diameter

dp/dx :

Streamwise pressure gradient

ES:

Effective slope = \(1{/}L_{s} \int {|\partial h^{\prime } {/}\partial x|} \partial x\)

Fl:

Flatness = \(1{/}N\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 4} } /\left[ {1{/}N\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 2} } } \right]^{2}\)

h :

Channel half-height = H/2

\(\overline{h}\) :

Mean roughness elevation

\(h^{\prime }\) :

Variation in roughness elevation about the mean

H :

Channel height

k :

Roughness height

k a :

Centerline average roughness height = \({1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime } }\)

k p :

Elevation of a roughness peak

k rms :

Root-mean-square roughness height = \(\sqrt {{1 \mathord{\left/ {\vphantom {1 {N\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 2} } }}} \right. \kern-\nulldelimiterspace} {N\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 2} } }}}\)

k s :

Equivalent sandgrain roughness height

k t :

Peak-to-trough roughness height = zmax − zmin

k v :

Elevation of a roughness valley

k z :

2N-point (N = 5) average peak-to-trough roughness height = \({1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}\sum\nolimits_{i = 1}^{N} {\left( {k_{pi} - k_{vi} } \right)}\)

L :

Channel length

L s :

Sampling length

Sk:

Skewness = \({{{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 3} } } \mathord{\left/ {\vphantom {{{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 3} } } {\left[ {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 2} } } \right]^{3/2} }}} \right. \kern-\nulldelimiterspace} {\left[ {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}\sum\nolimits_{i = 1}^{N} {h_{i}^{\prime 2} } } \right]^{3/2} }}\)

Rem :

Reynolds number based on channel height and bulk mean velocity = \(\overline{U}H/v\)

Reτ :

Friction Reynolds number =  \(U_{\tau } h/v\)

U :

Mean streamwise velocity

U CL :

Mean streamwise velocity at the channel centerline

\(\overline{U}\) :

Bulk mean velocity

U τ :

Friction velocity = \(\sqrt {{{\tau_{w} } \mathord{\left/ {\vphantom {{\tau_{w} } \rho }} \right. \kern-\nulldelimiterspace} \rho }}\)

W :

Channel width

y :

Distance from the wall

ΔU + :

Roughness function

\(\overline{{u^{\prime 2} }}\) :

Streamwise Reynolds normal stress

\(\overline{{v^{\prime 2} }}\) :

Wall-normal Reynolds normal stress

\(-\overline{{u^{\prime } v^{\prime } }}\) :

Reynolds shear stress

λ f :

Frontal solidity

λ p :

Plan solidity

Λ s :

Roughness density

δ :

Boundary-layer thickness

κ:

Von Kármán constant = 0.384

ν :

Kinematic viscosity of the fluid

ρ :

Density of the fluid

τ w :

Wall shear stress

+:

Inner variable (normalized with Uτ or ν/Uτ)

R :

Rough

S :

Smooth

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Acknowledgments

The authors would like to acknowledge the U.S. Office of Naval Research for financial support of this work under award numbers N0001412WX20078 (program manager Ronald Joslin) N0001422WX01363 (program manager Peter Chang). The assistance of the Naval Academy Hydromechanics Lab and the Project Support Branch is also gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Engineering, United States Naval Academy, Annapolis, MD, USA

    Karen A. Flack

  2. Department of Naval Architecture and Ocean Engineering, United States Naval Academy, Annapolis, MD, USA

    Michael P. Schultz

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  1. Karen A. Flack
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  2. Michael P. Schultz
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KAF and MPS are planned and conducted the experiments, analyzed the data, and wrote the manuscript.

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Correspondence to Michael P. Schultz.

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Flack, K.A., Schultz, M.P. Hydraulic characterization of sandpaper roughness. Exp Fluids 64, 3 (2023). https://doi.org/10.1007/s00348-022-03544-0

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