Direct numerical simulations of turbulent flow in a channel with superhydrophobic surfaces (SHS) were performed, and the effects of the surface texture on the turbulence and skin-friction coefficient were examined. The SHS is modeled as a planar boundary comprised of spanwise-alternating regions of no-slip and free-slip boundary conditions. Relative to the reference no-slip channel flow at the same bulk Reynolds number, the overall mean skin-friction coefficient is reduced by 21.6%. A detailed analysis of the turbulence kinetic energy budget demonstrates a reduction in production over the no-slip phases, which is explained by aid of quadrant analysis of the Reynolds shear stresses and statistical analysis of the turbulence structures. The results demonstrate a significant reduction in the strength of streamwise vortical structures in the presence of the SHS texture and a decrease in the Reynolds shear-stress component ⟨R12⟩ which has a favorable influence on drag over the no-slip phases. A secondary flow which is set up at the edges of the texture also effects a beneficial change in drag. Nonetheless, the skin-friction coefficient on the no-slip features is higher than the reference levels in a simple no-slip channel flow. The increase in the skin-friction coefficient is attributed to two factors. First, spanwise diffusion of the mean momentum from free-slip to no-slip regions increases the local skin-friction coefficient on the edges of the no-slip features. Second, the drag-reducing capacity of the SHS is further reduced due to additional Reynolds stresses, ⟨R13⟩.

The roughness functions induced by irregular peak- and/or pit-dominated surfaces in a fully developed turbulent channel flow are studied by direct numerical simulation. A surface generation algorithm is used to synthesise an irregular Gaussian height map with periodic boundaries. The Gaussian height map is decomposed into ‘pits-only’ and ‘peaks-only’ components, which produces two additional surfaces with similar statistical properties, with the exception of skewness, which are equal and opposite $({\mathcal{S}}=\pm 1.6)$. While the peaks-only surface yields a roughness function comparable to that of the Gaussian surface, the pits-only surface exhibits a far weaker roughness effect. Analysis of results is aided by deriving an equation for the roughness function that quantitatively identifies the mechanisms of momentum loss and/or gain. The statistical contributions of ‘form-induced’ and stochastic fluid motions to the roughness function are examined in further detail using quadrant analyses. Above the Gaussian and peaks-only surfaces, the contributions of dispersive and Reynolds shear stresses show a compensating effect, whereas above the pits-only surface, an additive effect is observed. Overall, the results emphasise the sensitivity of the near-wall flow with respect to higher-order topographical parameters, which can, in turn, induce significant differences in the roughness function above a peak- and/or pit-dominated surface.

In turbulent flows over streamwise-aligned superhydrophobic surface (SHS) textures, the percent drag reduction is dependent on Reynolds number. This dependence is examined using direct numerical simulations of channel flow over SHS texture at three bulk Reynolds numbers, Re b = 2800, 6785 and 10975.Simulations of regular no-slip channel flows at the same bulk velocities are also performed for comparison. Changes in the flow due to the SHS texture are examined with particular focus on phase averaged statistics and coherent turbulent motions. As the Reynolds number is increased, the texture performance, or the percent drag reduction, is enhanced. In terms of turbulent motions, the weakened The enhanced performance is therefore due to a significant drop in the strength of small-and large-scale vortical structures. The weaker vortical motions yield a considerable reduction in the Reynolds shear stress and the transport of mean momentum. The SHS texture introduces a performance penalty in the form of a local increase in drag at the no-slip free-slip edge. This adverse effect is caused by the inhomogeneity of the near-wall flow field. However, the affected region narrows in width at higher Reynolds numbers. As a result, the percent drag reduction approaches the gas fraction of the texture.

The influence of surface anisotropy upon the near-wall region of a rough-wall turbulent channel flow is investigated using direct numerical simulation (DNS). A set of nine irregular rough surfaces with fixed mean peak-to-valley height, near-Gaussian height distributions and specified streamwise and spanwise correlation lengths were synthesised using a surface generation algorithm. By defining the surface anisotropy ratio (SAR) as the ratio of the streamwise and spanwise correlation lengths of the surface, we demonstrate that surfaces with a strong spanwise anisotropy (SAR <1) can induce an over 200% increase in the roughness function ΔU + , compared to their streamwise anisotropic (SAR >1) equivalent. Furthermore, we find that the relationship between the roughness function ΔU + and the SAR parameter approximately follows an exponentially decaying function. The statistical response of the near-wall flow is studied using a "double-averaging" methodology in order to distinguish form-induced "dispersive" stresses from their turbulent counterparts. Outer-layer similarity is recovered for the mean velocity defect profile as well as the Reynolds stresses. The dispersive stresses all attain their maxima within the roughness canopy. Only the streamwise dispersive stress reaches levels that are comparable to the equivalent Reynolds stress, with surfaces of high SAR attaining the highest levels of streamwise dispersive stress. The Reynolds stress anisotropy also shows distinct differences between cases with strong streamwise anisotropy that stay close to an axisymmetric, rodlike state for all wall-normal locations, compared to cases with spanwise anisotropy where an axisymmetric, disk-like state of the Reynolds stress anisotropy tensor is observed around the roughness mean plane. Overall, the results from this study underline that the drag penalty incurred by a rough surface is strongly influenced by the surface topography and highlight its impact upon the mean momentum deficit in the outer flow as well as the Reynolds and dispersive stresses within the roughness layer.

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Whereas streamwise effective slope ( $ES_{x}$ ) is accepted as a key topographical parameter in the context of rough-wall turbulent flows, the significance of its spanwise counterpart ( $ES_{y}$ ) remains largely unexplored. Here, the response of turbulent channel flow over irregular, three-dimensional rough walls with systematically varied values of $ES_{y}$ is studied using direct numerical simulation. All simulations were performed at a fixed friction Reynolds number 395, corresponding to a viscous-scaled roughness height $k^{+}\approx 65.8$ (where $k$ is the mean peak-to-valley height). A surface generation algorithm is used to synthesise a set of ten irregular surfaces with specified $ES_{y}$ for three different values of $ES_{x}$ . All surfaces share a common mean peak-to-valley height and are near-Gaussian, which allows this study to focus on the impact of varying $ES_{y}$ , since roughness amplitude, skewness and $ES_{x}$ can be eliminated simultaneously as parameters. Based on an analysis of first- and second-order velocity statistics, as well as turbulence co-spectra and the fractional contribution of pressure and viscous drag, the study shows that $ES_{y}$ can strongly affect the roughness drag penalty – particularly for low- $ES_{x}$ surfaces. A secondary observation is that particular low- $ES_{y}$ surfaces in this study can lead to diminished levels of outer-layer similarity in both mean flow and turbulence statistics, which is attributed to insufficient scale separation between the outer length scale and the in-plane spanwise roughness wavelength.