Reliable full-scale prediction of drag due to rough wall-bounded turbulent fluid flow remains a challenge. Currently, the uncertainty is at least 10%, with consequences, for example, on energy and transport applications exceeding billions of dollars per year. The crux of the difficulty is the large number of relevant roughness topographies and the high cost of testing each topography, but computational and experimental advances in the last decade or so have been lowering these barriers. In light of these advances, here we review the underpinnings and limits of relationships between roughness topography and drag behavior, focusing on canonical and fully turbulent incompressible flow over rigid roughness. These advances are beginning to spill over into multiphysical areas of roughness, such as heat transfer, and promise broad increases in predictive reliability.

A review of predictive methods used to determine the frictional drag on a rough surface is presented. These methods utilize a wide range of roughness scales, including roughness height, pitch, density, and shape parameters. Most of these scales were developed for regular roughness, limiting their applicability to predict the drag for many engineering flows. A new correlation is proposed to estimate the frictional drag for a surface covered with three-dimensional, irregular roughness in the fully rough regime. The correlation relies solely on a measurement of the surface roughness profile and builds on previous work utilizing moments of the surface statistics. A relationship is given for the equivalent sandgrain roughness height as a function of the root-mean-square roughness height and the skewness of the roughness probability density function. Boundary layer similarity scaling then allows the overall frictional drag coefficient to be determined as a function of the ratio of the equivalent sandgrain roughness height to length of the surface.

The Reynolds number similarity hypothesis of Townsend ͓The Structure of Turbulent Shear Flow ͑Cambridge University Press, Cambridge, UK, 1976͔͒ states that the turbulence beyond a few roughness heights from the wall is independent of the surface condition. The underlying assumption is that the boundary layer thickness ␦ is large compared to the roughness height k. This hypothesis was tested experimentally on two types of three-dimensional rough surfaces. Boundary layer measurements were made on flat plates covered with sand grain and woven mesh roughness in a closed return water tunnel at a momentum thickness Reynolds number Re of ϳ14 000. The boundary layers on the rough walls were in the fully rough flow regime ͑k s + ജ 100͒ with the ratio of the boundary layer thickness to the equivalent sand roughness height ␦ / k s greater than 40. The results show that the mean velocity profiles for rough and smooth walls collapse well in velocity defect form in the overlap and outer regions of the boundary layer. The Reynolds stresses for the two rough surfaces agree well throughout most of the boundary layer and collapse with smooth wall results outside of 3k s. Higher moment turbulence statistics and quadrant analysis also indicate the differences in the rough wall boundary layers are confined to y Ͻ 5k s. The present results provide support for Townsend's Reynolds number similarity hypothesis for uniform three-dimensional roughness in flows where ␦ / k s ജ 40.

Turbulence measurements for rough-wall boundary layers are presented and compared to those for a smooth wall. The rough-wall experiments were made on a three-dimensional rough surface geometrically similar to the honed pipe roughness used by Shockling, Allen & Smits (J. Fluid Mech. vol. 564, 2006, p. 267). The present work covers a wide Reynolds-number range (Reθ = 2180–27 100), spanning the hydraulically smooth to the fully rough flow regimes for a single surface, while maintaining a roughness height that is a small fraction of the boundary-layer thickness. In this investigation, the root-mean-square roughness height was at least three orders of magnitude smaller than the boundary-layer thickness, and the Kármán number (δ+), typifying the ratio of the largest to the smallest turbulent scales in the flow, was as high as 10100. The mean velocity profiles for the rough and smooth walls show remarkable similarity in the outer layer using velocity-defect scaling. The Reynolds stresses and higher-order turbulence statistics also show excellent agreement in the outer layer. The results lend strong support to the concept of outer layer similarity for rough walls in which there is a large separation between the roughness length scale and the largest turbulence scales in the flow.

This paper outlines the authors' experimental research in rough-wall-bounded turbulent flows that has spanned the past 15 years. The results show that, in general, roughness effects are confined to the inner layer. In accordance with Townsend's Reynolds number similarity hypothesis, the outer layer is insensitive to surface condition except in the role it plays in setting the length and velocity scales for the outer flow. An exception to this can be two-dimensional roughness which has been observed in some cases to suffer roughness effects far from the wall. However, recent results indicate that similarity also holds for two-dimensional roughness provided the Reynolds number is large, and there is sufficient scale separation between the roughness length scale and the boundary layer thickness. The concept of similarity between smooth-and rough-wall flows is of great practical importance as most computational and analytical modeling tools rely on it either explicitly or implicitly in predicting flows over rough walls. Because of the observed similarity, the roughness function (U +), or shift in the log layer, is a useful way of characterizing the roughness effect on the mean flow and the frictional drag. In the fully rough regime, it is shown that the hydraulic roughness length scale is related to the root-mean-square height (k rms) and skewness (s k) of the surface elevation probability density function. On the other hand, the onset of roughness effects is seen to be associated with the largest surface features which are typified by the peak-to-trough height (k t). Roughness function behavior in the transitionally rough regime varies significantly between roughness types. Since no "universal" roughness function exists, no single roughness length scale can characterize all roughness types in all the flow regimes. Despite this, research using roughness with a systematic variation in texture is ongoing in an effort to uncover surface parameters that lead to the variation in the frictional drag behavior witnessed in the transitionally rough regime. [

The existence of a critical roughness height for outer layer similarity between smooth and rough wall turbulent boundary layers is investigated. Results are presented for boundary layer measurements on flat plates covered with sandgrain and woven mesh with the ratio of the boundary layer thickness to roughness height ͑␦ / k͒ varying from 16 to 110 at Re = 7.3ϫ 10 3-13ϫ 10 3. In all cases tested, the layer directly modified by the roughness ͑the roughness sublayer͒ is confined to a region Ͻ5k or Ͻ3k s from the wall ͑where k s is the equivalent sandgrain roughness height͒. In the larger roughness cases, this region of turbulence modification extends into the outer flow. However, beyond 5k or 3k s from the wall, similarity in the turbulence quantities is observed between the smooth and rough wall boundary layers. These results indicate that a critical roughness height, where the roughness begins to affect most or all of the boundary layer, does not exist. Instead, the outer flow is only gradually modified with increasing roughness height as the roughness sublayer begins to occupy an ever increasing fraction of the outer layer.

Turbulence measurements for rough-wall boundary layers are presented and compared to those for a smooth wall. The rough-wall experiments were made on a woven mesh surface at Reynolds numbers approximately equal to those for the smooth wall. Fully rough conditions were achieved. The present work focuses on turbulence structure, as documented through spectra of the fluctuating velocity components, swirl strength, and two-point auto- and cross-correlations of the fluctuating velocity and swirl. The present results are in good agreement, both qualitatively and quantitatively, with the turbulence structure for smooth-wall boundary layers documented in the literature. The boundary layer is characterized by packets of hairpin vortices which induce low-speed regions with regular spanwise spacing. The same types of structure are observed for the rough- and smooth-wall flows. When the measured quantities are normalized using outer variables, some differences are observed, but quantitative similarity, in large part, holds. The present results support and help to explain the previously documented outer-region similarity in turbulence statistics between smooth- and rough-wall boundary layers.

Results of an experimental investigation of the flow over a model roughness are presented. The series of roughness consists of close-packed pyramids in which both the height and the slope were systematically varied. The aim of this work was to document the mean flow and subsequently gain insight into the physical roughness scales which contribute to drag. The mean velocity profiles for all nine rough surfaces collapse with smooth-wall results when presented in velocity-defect form, supporting the use of similarity methods. The results for the six steepest surfaces indicate that the roughness function ⌬U + scales almost entirely on the roughness height with little dependence on the slope of the pyramids. However, ⌬U + for the three surfaces with the smallest slope does not scale satisfactorily on the roughness height, indicating that these surfaces might not be thought of as surface "roughness" in a traditional sense but instead surface "waviness."