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Research Papers: Fundamental Issues and Canonical Flows

# Toward a Universal Roughness Correlation

[+] Author and Article Information
Pourya Forooghi

Institute of Fluid Mechanics,
Karlsruhe Institute of Technology,
Kaiserstraße 10,
Karlsruhe 76131, Germany
e-mail: forooghi@kit.edu

Alexander Stroh

Institute of Fluid Mechanics,
Karlsruhe Institute of Technology,
Kaiserstraße 10,
Karlsruhe 76131, Germany
e-mail: alexander.stroh@kit.edu

Franco Magagnato

Institute of Fluid Mechanics,
Karlsruhe Institute of Technology,
Kaiserstraße 10,
Karlsruhe 76131, Germany
e-mail: franco.magagnato@kit.edu

Suad Jakirlić

Professor
Institute of Fluid Mechanics and Aerodynamics,
Technical University Darmstadt,
Alarich-Weiss-Straße 10,
Darmstadt 64287, Germany
e-mail: jakirlic@sla.tu-darmstadt.de

Bettina Frohnapfel

Professor
Institute of Fluid Mechanics,
Karlsruhe Institute of Technology,
Kaiserstraße 10,
Karlsruhe 76131, Germany
e-mail: bettina.frohnapfel@kit.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 1, 2017; final manuscript received July 4, 2017; published online August 28, 2017. Assoc. Editor: Sergio Pirozzoli.

J. Fluids Eng 139(12), 121201 (Aug 28, 2017) (12 pages) Paper No: FE-17-1072; doi: 10.1115/1.4037280 History: Received February 01, 2017; Revised July 04, 2017

## Abstract

The effects of several surface parameters on equivalent sand roughness (ks) in fully rough regime are investigated by means of direct numerical simulation (DNS) of flow in channels with different wall geometries at $Reτ≅500$. The roughness geometry is generated by randomly distributing roughness elements of random size and prescribed shape on a flat surface. The roughness generation approach allows systematic variation of moments of surface height probability density function (PDF), size distribution of roughness peaks, and surface slope. A total number of 38 cases are solved. It is understood that a correlation based on surface height skewness and effective slope (ES) can satisfactorily predict ks normalized with maximum peak-to-valley roughness height within a major part of the studied parameter space. Such a correlation is developed based on the present data points and a number of complementary data points from the literature. It is also shown that the peak size distribution can independently influence the skin friction; at fixed values of rms surface height, skewness, kurtosis, and ES, a surface with uniform size peaks causes higher skin friction compared to one with nonuniform peak sizes. Additionally, it is understood that a roughness generated by regular arrangement of roughness elements may lead to a significantly different skin friction compared to a random arrangement. A staggered and an aligned regular arrangement are examined in this paper and it is observed that the former produces significantly closer results to the corresponding random arrangement.

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## Figures

Fig. 1

Calculation of wall shear stress for case A7088 as an example. Profile of total shear stress (thick solid line) is extrapolated (thin dashed line) to find the value at y = 0 and this value is defined as τw. Viscous and Reynolds stress parts of total shear stress are also plotted using thick dashed lines.

Fig. 2

Schematic representation of the roughness generation approach. As shown on the right-hand side, element profiles can be varied to adjust the surface statistics.

Fig. 3

Square patches of selected surface samples. All dimensions are normalized with effective channel half-height h. Coloring (grayscale in the printed version) indicates surface elevation. First row: A7088 (reference sample); second row from left to right: B7088 and C7088 (samples with the same ES and Δ as the reference sample but different Sk); third row from left to right right: A3588 and A0088 (samples with the same ES and Sk as the reference sample but different Δ); fourth row from left to right: A7060 and A7040 (samples with the same Sk and Δ as the reference sample but different ES).

Fig. 4

Square patches of the three samples discussed in Sec. 4.1.1. Up: D0088; bottom from left to right: D0088s and D0088a. The two samples below have identical elements as the one above but in different arrangements. Coloring (grayscale in the printed version) is similar to Fig. 3. The mean flow direction is from bottom left to top right.

Fig. 5

Equivalent sand roughness (top), mean velocity defect profiles in outer scale (middle), and rms fluctuating velocities (bottom) for the three cases discussed in Sec. 4.1.1. For clarity, in the last figure, the profiles of u and v are shifted upward by 1.5 and 0.75 units, respectively.

Fig. 6

Mean velocity defect profiles in the outer scale for the cases discussed in Sec. 4.1.2

Fig. 7

Variation of ks normalized with krms (top) and kz (bottom) against Δ for the cases discussed in Sec. 4.1.2. In all cases, ES is 0.88. Each color indicates on a value of Δ; black: Δ = 0.7, blue (dark gray in the printed version): Δ = 0.35, orange (light gray): Δ = 0.15, and red (middle gray): Δ = 0. Each symbol type indicates on a value of Sk; down-triangle: Sk = − 0.33, circle: Sk = 0.21, and triangle: Sk = 0.67. In all cases, Kurtosis is 2.61, except those with hollow symbols in which Ku = 1.9.

Fig. 8

Variation of ks normalized with krms (top) and kz (bottom) against Sk for the cases discussed in Sec. 4.1.2. Symbols are same as in Fig. 7. Top: cross symbols show cases with lower values of ES. Dashed line is the Flack and Schultz correlation (Eq. (4)). Bottom: solid line is the function F(Sk) (Eq.(11)).

Fig. 9

Variation of ks normalized with kp against density parameter Λs for all cases. Dashed and dotted–dashed lines are roughness correlations due to Sigal and Danberg (Eq. (7)) and van Rij (Eq. (8)), respectively. Symbols are same as in Fig. 7.

Fig. 10

Variation of ks normalized with krms (top), kz·F(Sk) (middle), and kz·F̃(Sk,Δ) (bottom) against ES. The solid lines in the second and third graphs are G(ES) and G̃(ES) used in Eqs. (13) and (14), respectively. Cross and plus symbols show data from Yuan and Piomelli [11] and Thakkar et al. [22], respectively. Other symbols are from the present simulations and have the same meaning as in Fig. 7. There is a ±15% uncertainty interval associated with the cross symbols.

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