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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, $\Lambda$ (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, $\Delta U^+$. 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 ($\Lambda \lesssim 0.15$) 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 $\Lambda\gtrsim0.15-$0.18, in agreement with the literature. A model is proposed for the limit of $\Lambda\rightarrow\infty$, 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 increasingly further away from the wall with increasing solidity. In the sparse regime, increasing $\Lambda$ reduces streamwise turbulent energy associated with the near-wall cycle, while increasing $\Lambda$ in the dense regime increases turbulent energy [continued]$\ldots$, Comment: Accepted in the Journal of Fluid Mechanics

We conduct minimal-channel direct numerical simulations of turbulent flow over two-dimensional rectangular bars aligned in the spanwise direction. This roughness has been often described as $d$-type, as the roughness function $\Delta U^+$ is thought to depend only on the outer-layer length scale (pipe diameter, channel half height or boundary layer thickness). This is in contrast to conventional engineering rough surfaces, named $k$-type, for which $\Delta U^+$ depends on the roughness height, $k$. The minimal-span rough-wall channel is used to circumvent the high cost of simulating high Reynolds number flows, enabling a range of bars with varying aspect ratios to be investigated. The present results show that increasing the trough-to-crest height ($k$) of the roughness while keeping the width between roughness bars, $\mathcal{W}$, fixed in viscous units, results in non-$k$-type behaviour although this does not necessarily indicate $d$-type behaviour. Instead, for deep surfaces with $k/\mathcal{W}\gtrsim 3$, the roughness function appears to depend only on $\mathcal{W}$ in viscous units. In these situations, the flow no longer has any information about how deep the roughness is and instead can only `see' the width of the fluid gap between the bars., Comment: Accepted in the Journal of Fluid Mechanics

We conducted direct numerical simulations (DNSs) of turbulent flow over three-dimensional sinusoidal roughness in a channel. A passive scalar is present in the flow with Prandtl number $Pr=0.7$, to study heat transfer by forced convection over this rough surface. The minimal channel is used to circumvent the high cost of simulating high Reynolds number flows, which enables a range of rough surfaces to be efficiently simulated. The near-wall temperature profile in the minimal channel agrees well with that of the conventional full-span channel, indicating it can be readily used for heat-transfer studies at a much reduced cost compared to conventional DNS. As the roughness Reynolds number, $k^+$, is increased, the Hama roughness function, $\Delta U^+$, increases in the transitionally rough regime before tending towards the fully rough asymptote of $\kappa_m^{-1}\log(k^+)+C$, where $C$ is a constant that depends on the particular roughness geometry and $\kappa_m\approx0.4$ is the von K\'arm\'an constant. In this fully rough regime, the skin-friction coefficient is constant with bulk Reynolds number, $Re_b$. Meanwhile, the temperature difference between smooth- and rough-wall flows, $\Delta\Theta^+$, appears to tend towards a constant value, $\Delta\Theta^+_{FR}$. This corresponds to the Stanton number (the temperature analogue of the skin-friction coefficient) monotonically decreasing with $Re_b$ in the fully rough regime. Using shifted logarithmic velocity and temperature profiles, the heat transfer law as described by the Stanton number in the fully rough regime can be derived once both the equivalent sand-grain roughness $k_s/k$ and the temperature difference $\Delta \Theta^+_{FR}$ are known. In meteorology, this corresponds to the ratio of momentum and heat transfer roughness lengths, $z_{0m}/z_{0h}$, being linearly proportional to $z_{0m}^+$, the momentum roughness length [continued]..., Comment: Accepted (In press) in the Journal of Fluid Mechanics

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar. 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. 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 to reduce the high streamwise velocity. 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. 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. Good agreement with the literature for the same roughness geometry is observed., Comment: Published in the Journal of Fluid Mechanics

This fluid dynamics video submitted to the Gallery of Fluid motion shows a turbulent boundary layer developing under a 5 metre-long flat plate towed through water. A stationary imaging system provides a unique view of the developing boundary layer as it would form over the hull of a ship or fuselage of an aircraft. The towed plate permits visualisation of the zero-pressure-gradient turbulent boundary layer as it develops from the trip to a high Reynolds number state ($Re_\tau \approx 3000$). An evolving large-scale coherent structure will appear almost stationary in this frame of reference. The visualisations provide an unique view of the evolution of fundamental processes in the boundary layer (such as interfacial bulging, entrainment, vortical motions, etc.). In the more traditional laboratory frame of reference, in which fluid passes over a stationary body, it is difficult to observe the full evolution and lifetime of turbulent coherent structures. An equivalent experiment in a wind/water-tunnel would require a camera and laser that moves with the flow, effectively `chasing' eddies as they advect downstream., Comment: This submission contains two video files for the Gallery of Fluid Motion