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A theoretically based relationship for the Darcy–Weisbach friction factor $f$ for rough-bed open-channel flows is derived and discussed. The derivation procedure is based on the double averaging (in time and space) of the Navier–Stokes equation followed by repeated integration across the flow. The obtained relationship explicitly shows that the friction factor can be split into at least five additive components, due to: (i) viscous stress; (ii) turbulent stress; (iii) dispersive stress (which in turn can be subdivided into two parts, due to bed roughness and secondary currents); (iv) flow unsteadiness and non-uniformity; and (v) spatial heterogeneity of fluid stresses in a bed-parallel plane. These constitutive components account for the roughness geometry effect and highlight the significance of the turbulent and dispersive stresses in the near-bed region where their values are largest. To explore the potential of the proposed relationship, an extensive data set has been assembled by employing specially designed large-eddy simulations and laboratory experiments for a wide range of Reynolds numbers. Flows over self-affine rough boundaries, which are representative of natural and man-made surfaces, are considered. The data analysis focuses on the effects of roughness geometry (i.e. spectral slope in the bed elevation spectra), relative submergence of roughness elements and flow and roughness Reynolds numbers, all of which are found to be substantial. It is revealed that at sufficiently high Reynolds numbers the roughness-induced and secondary-currents-induced dispersive stresses may play significant roles in generating bed friction, complementing the dominant turbulent stress contribution.

Knowledge of hydraulic resistance of single-valued self-affine fractal surfaces remains very limited. To advance this area, a set of experiments have been conducted in two separate open-channel flumes to investigate the effects of the spectral structure of bed roughness on the drag at the bed. Three self-affine fractal roughness patterns, based on a simple but realistic three-range spectral model, have been investigated with spectral scaling exponents of − 1, − 5/3 and − 3, respectively. The different widths of the flumes and a range of flow depths also afforded an opportunity to consider effects of the flow aspect ratio and relative submergence. The results show that with all else equal the friction factor increases as the spectral exponent decreases. In addition, the relationship between the spectral exponent and effective slope of the roughness is demonstrated, for the first time. Aspect ratio effects on the friction factor within the studied range were found to be negligible.

Abstract. In natural open-channel flows over complex surfaces, a wide range of superimposed roughness elements may contribute to flow resistance. Gravel-bed rivers present a particularly interesting example of this kind of multiscalar flow resistance problem, as both individual grains and bedforms may contribute to the roughness length. In this paper, we propose a novel method of estimating the relative contribution of different physical scales of in-channel topography to the total roughness length, using a transform-roughness correlation (TRC) approach. The technique, which uses a longitudinal profile, consists of (1) a wavelet transform which decomposes the surface into roughness elements occurring at different wavelengths and (2) a “roughness correlation” that estimates the roughness length (ks) associated with each wavelength based on its geometry alone. When applied to original and published laboratory experiments with a range of channel morphologies, the roughness correlation estimates the total ks to approximately a factor of 2 of measured values but may perform poorly in very steep channels with low relative submergence. The TRC approach provides novel and detailed information regarding the interaction between surface topography and fluid dynamics that may contribute to advances in hydraulics, bedload transport, and channel morphodynamics.

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