Direct numerical simulations (DNS) are conducted for turbulent flow through pipes with three-dimensional sinusoidal roughnesses explicitly represented by body-conforming grids. The same viscous-scaled roughness geometry is first simulated at a range of different Reynolds numbers to investigate the effects of low Reynolds numbers and low R 0 /h, where R 0 is the pipe radius and h is the roughness height. Results for the present class of surfaces show that the Hama roughness function U + is only marginally affected by low Reynolds numbers (or low R 0 /h), and observations of outer-layer similarity (or lack thereof) show no signs of sensitivity to Reynolds number. Then, building on this, a systematic approach is taken to isolate the effects of roughness height h + and wavelength λ + in a turbulent wall-bounded flow in both transitionally rough and fully rough regimes. Current findings show that while the effective slope ES (which for the present sinusoidal surfaces is proportional to h + /λ + ) is an important roughness parameter, the roughness function U + must also depend on some measure of the viscous roughness height. A simplistic linear-log fit clearly illustrates the strong correlation between U + and both the roughness average height k + a (which is related to h + ) and ES for the surfaces simulated here, consistent with published literature. Various definitions of the virtual origin for rough-wall turbulent pipe flow are investigated and, for the surfaces simulated here, the hydraulic radius of the pipe appears to be the most suitable parameter, and indeed is the only virtual origin that can ever lead to collapse in the total stress. First-and second-order statistics are also analysed and collapses in the outer layer are observed for all cases, including those where the largest roughness height is a substantial proportion of the reference radius (low R 0 /h). These results provide evidence that turbulent pipe flow over the present sinusoidal surfaces adheres to Townsend's notion of outer-layer similarity, which pertains to statistics of relative motion.

The occurrence of secondary flows is investigated for three-dimensional sinusoidal roughness where the wavelength and height of the roughness elements are systematically altered. The flow spanned from the transitionally rough regime up to the fully rough regime and the solidity of the roughness ranged from a wavy, sparse roughness to a dense roughness. Analysing the time-averaged velocity, secondary flows are observed in all of the cases, reflected in the coherent stress profile which is dominant in the vicinity of the roughness elements. The roughness sublayer, defined as the region where the coherent stress is non-zero, scales with the roughness wavelength when the roughness is geometrically scaled (proportional increase in both roughness height and wavelength) and when the wavelength increases at fixed roughness height. Premultiplied energy spectra of the streamwise velocity turbulent fluctuations show that energy is reorganised from the largest streamwise wavelengths to the shorter streamwise wavelengths. The peaks in the premultiplied spectra at the streamwise and spanwise wavelengths are correlated with the roughness wavelength in the fully rough regime. Current simulations show that the spanwise scale of roughness determines the occurrence of large-scale secondary flows.

We describe a fast direct numerical simulation (DNS) method that promises to directly characterise the hydraulic roughness of any given rough surface, from the hydraulically smooth to the fully rough regime. The method circumvents the unfavourable computational cost associated with simulating high-Reynolds-number flows by employing minimal-span channels (Jiménez & Moin, J. Fluid Mech., vol. 225, 1991, pp. 213-240). Proof-of-concept simulations demonstrate that flows in minimal-span channels are sufficient for capturing the downward velocity shift, that is, the Hama roughness function, predicted by flows in full-span channels. We consider two sets of simulations, first with modelled roughness imposed by body forces, and second with explicit roughness described by roughness-conforming grids. Owing to the minimal cost, we are able to conduct direct numerical simulations with increasing roughness Reynolds numbers while maintaining a fixed blockage ratio, as is typical in full-scale applications. The present method promises a practical, fast and accurate tool for characterising hydraulic resistance directly from profilometry data of rough surfaces.

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar with respect to the wall shear stress. This makes it a prime candidate for the minimal-span channel, which only captures the near-wall flow by restricting the spanwise channel width to be of the order of a few hundred viscous units. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418-431) showed that a minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following this, we aim to investigate the fundamental dynamics of the minimal-span channel framework with an eye towards further improving performance. The streamwise domain length of the channel is investigated with the minimum length found to be three times the spanwise width or 1000 viscous units, whichever is longer. The outer layer of the minimal channel is inherently unphysical and as such alterations to it can be performed so long as the near-wall flow, which is the same as in a full-span channel, remains unchanged. Firstly, a half-height (open) channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel. This reduces the high streamwise velocity associated with the minimal channel and allows for a larger computational time step. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface. Currently, multiple steady simulations at fixed roughness Reynolds numbers are needed to obtain this behaviour. The results indicate that the non-dimensional pressure gradient parameter must be kept below 0.03-0.07 to ensure that pressure gradient effects do not lead to an inaccurate roughness function. An empirical costing argument is developed to determine the cost in terms of CPU hours of minimal-span channel simulations a priori. This argument involves counting the number of eddy lifespans in the channel, which is then related to the statistical uncertainty of the streamwise velocity. For a given statistical uncertainty in the roughness function, this can then be used to determine the simulation run time. Following this, a finite-volume code with a body-fitted grid is used to determine the roughness function for square-based pyramids using the above insights. Comparisons to experimental studies for the same roughness geometry are made and good agreement is observed.

We investigate rough-wall turbulent flows through direct numerical simulations of flow over three-dimensional transitionally rough sinusoidal surfaces. The roughness Reynolds number is fixed at $k^{+}=10$, where $k$ is the sinusoidal semi-amplitude, and the sinusoidal wavelength is varied, resulting in the roughness solidity $\unicode[STIX]{x1D6EC}$ (frontal area divided by plan area) ranging from 0.05 to 0.54. The high cost of resolving both the flow around the dense roughness elements and the bulk flow is circumvented by the use of the minimal-span channel technique, recently demonstrated by Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) to accurately determine the Hama roughness function, $\unicode[STIX]{x0394}U^{+}$. Good agreement of the second-order statistics in the near-wall roughness-affected region between minimal- and full-span rough-wall channels is observed. In the sparse regime of roughness ($\unicode[STIX]{x1D6EC}\lesssim 0.15$) the roughness function increases with increasing solidity, while in the dense regime the roughness function decreases with increasing solidity. It was found that the dense regime begins when $\unicode[STIX]{x1D6EC}\gtrsim 0.15{-}0.18$, in agreement with the literature. A model is proposed for the limit of $\unicode[STIX]{x1D6EC}\rightarrow \infty$, which is a smooth wall located at the crest of the roughness elements. This model assists with interpreting the asymptotic behaviour of the roughness, and the rough-wall data presented in this paper show that the near-wall flow is tending towards this modelled limit. The peak streamwise turbulence intensity, which is associated with the turbulent near-wall cycle, is seen to move further away from the wall with increasing solidity. In the sparse regime, increasing $\unicode[STIX]{x1D6EC}$ reduces the streamwise turbulent energy associated with the near-wall cycle, while increasing $\unicode[STIX]{x1D6EC}$ in the dense regime increases turbulent energy. An analysis of the difference of the integrated mean momentum balance between smooth- and rough-wall flows reveals that the roughness function decreases in the dense regime due to a reduction in the Reynolds shear stress. This is predominantly due to the near-wall cycle being pushed away from the roughness elements, which leads to a reduction in turbulent energy in the region previously occupied by these events.