The description of riblets and other drag-reducing devices has long used the concept of longitudinal and transverse protrusion heights, both as a means to predict the drag reduction itself and as equivalent boundary conditions to simplify numerical simulations by transferring the effect of riblets onto a flat virtual boundary. The limitation of this idea is that it stems from a first-order approximation in the riblet-size parameter s^+, and as a consequence it cannot predict other than a linear dependence of drag reduction upon s^+; in other words, the initial slope of the drag-reduction curve. Here the concept is extended to a full asymptotic expansion using matched asymptotics, which consistently provides higher-order protrusion heights and higher-order equivalent boundary conditions on a virtual flat surface. This procedure also allows us to explore nonlinearities of the Navier-Stokes equations and the way they enter the s^+-expansion, with somewhat surprising results.

This study investigates the spatial development and recovery of near-wall flow steering induced by yawed widely-spaced riblets. To accurately measure the mean spanwise velocity in the near-wall region, we developed a unique, customised rotating probe equipped with a single hot-wire. This novel technique demonstrates high sensitivity, reliably detecting small spanwise velocity components. We applied this probe to map the flow over straight-to-yawed and yawed-to-straight riblet configurations. For the straight-to-yawed transition, the downstream distribution of the mean spanwise velocity closely aligns with the step-forced spatial Stokes layer (SSL) solution, achieving an equivalent active wall motion of V^+_equ≈ 2.0. This offers a simplified analytical pathway for future passive flow manipulation studies. Conversely, flow recovery over the yawed-to-straight configuration diverges from the SSL step-down phase prediction. Instead, a spatial lag was observed where crest-adjacent flow recovered quickly, but significant residual spanwise flow (V^+ > 0.5) persisted far downstream (x^+ > 1000).

The formation of Kelvin–Helmholtz-like rollers (referred to as K–H rollers) over riblet surfaces has been linked to the drag-increasing behaviour seen in certain riblet geometries, such as sawtooth and blade riblets, when the riblet size reaches sufficiently large viscous scales (Endrikat et al. (2021 a ), J. Fluid Mech . 913 , A37). In this study, we focus on the sawtooth geometry of fixed physical size, and experimentally examine the response of these K–H rollers to further increases in viscous scaled riblet sizes, by adopting the conventional approach of increasing freestream speeds (and consequently, the friction Reynolds number). Rather than continual strengthening, the present study shows a gradual weakening of these K–H rollers with increasing sawtooth riblet size. This is achieved by an analysis of the roller geometric characteristics using both direct numerical simulations and hot-wire anemometry databases at matched viscous scaled riblet spacings, with the former used to develop a novel methodology for detecting these rollers via streamwise velocity signatures (e.g. as acquired by hot wires). Spectral analysis of the streamwise velocity time series, acquired within riblet grooves, reveals that the frequencies (and the streamwise wavelengths) of the K–H rollers increase with increasing riblet size. Cross-correlation spectra, estimated from unique two-point hot-wire measurements in the cross-plane, show a weakening of the K–H rollers and a reduction in their wall-normal coherence with increasing riblet size. Besides contributing to our understanding of the riblet drag-increasing mechanisms, the present findings also have implications for the heat transfer enhancing capabilities of sawtooth riblets, which have been associated previously with the formation of K–H rollers. The present study also suggests conducting future investigations by decoupling the effects of viscous scaled riblet spacing and friction Reynolds numbers, to characterise their influence on the K–H rollers independently.

Wall reflection of laser light in particle image velocimetry (PIV) measurements is one of the limiting factors in obtaining velocity information in the vicinity of rough walls. In this wind tunnel study, wall reflections for stereoscopic PIV (SPIV) measurements over a rough wall are suppressed by coating the rough surface with fluorescent paint to shift the diffuse-reflected light to a higher wavelength. A narrow band-pass filter employed on the imaging lenses extinguishes the majority of this wavelength-shifted surface-reflected laser light, allowing particle images to be recorded in the near-wall vicinity. Three different fluorescent paints were evaluated. Rhodamine 6G was found to have the best performance for suppressing wall and background reflections. Reliable velocity measurements with this technique were obtained as close as 500 µm ( ∼ 5.0 wall units) from the base of the roughness valleys. The technique was successfully used with SPIV to detect the secondary motions within riblet valleys.

We investigate nonlinear energy transfer for channel flows at friction Reynolds numbers $Re_{\tau }=180$ and $590$ . The key feature of the analysis is that we quantify the energy transferred from a source mode to a recipient mode, with each mode characterised by a streamwise wavenumber and a spanwise wavenumber. This is achieved through an explicit examination of the triadic interactions of the nonlinear energy transfer term in the spectral turbulent kinetic energy equation. First, we quantify the nonlinear energy transfer gain and loss for individual Fourier modes. The gain and loss cannot be obtained without expanding the nonlinear triadic interactions. Second, we quantify the nonlinear energy transfer budgets for three types of modes. Each type of mode is characterised by a specific region in streamwise–spanwise wavenumber space. We find that a transverse cascade from streamwise-elongated modes to spanwise-elongated modes exists for all three types of modes. Third, we quantify the forward and inverse cascades between resolved scales and subgrid scales in the spirit of large-eddy simulations. For the cutoff wavelength range that we consider, the forward and inverse cascades between the resolved scales and subgrid scales result in a net forward cascade from the resolved scales to the subgrid scales. The shape of the net forward cascade curve with respect to the cutoff wavelength resembles the net forward cascade predicted by the Smagorinsky eddy viscosity.

To aid in prediction of turbulent boundary layer flows over rough surfaces, a new model is proposed to estimate hydrodynamic roughness based solely on geometric surface information. The model is based on a fluid-mechanics motivated geometric parameter called the wind-shade factor. Sheltering is included using a rapid algorithm adapted from the landscape shadow literature, while local pressure drag is estimated using a piecewise potential flow approximation. Similarly to evaluating traditional surface parameters such as skewness or average slope magnitude, the wind-shade factor is purely geometric and can be evaluated efficiently from knowing the surface elevation map and the mean flow direction. The wind-shade roughness model is applied to over 100 different surfaces available in a public roughness database and some others, and the predicted sandgrain-roughness heights are compared with measured values. Effects of various model ingredients are analysed, and transitionally rough surfaces are treated by adding a term representing the viscous stress component.

By examining a systematic set of direct numerical simulations, we develop a model which captures the effect of roughness density on global and local heat transfer in forced convection. The surfaces considered are zero-skewed three-dimensional sinusoidal rough walls with solidities, $\varLambda$ (defined as the frontal area divided by the total plan area), ranging from low $\varLambda = 0.09$ , medium $\varLambda = 0.18$ to high $\varLambda = 0.36$ . For each solidity, we vary the roughness height characterised by the roughness Reynolds number, $k^+$ , from transitionally rough to fully rough conditions. The findings indicate that, as the fully rough regime is approached, there is a pronounced breakdown in the analogy between heat and momentum transfer, whereby the velocity roughness function $\Delta U^+$ continues to increase and the temperature roughness function $\Delta \varTheta ^+$ attains a peak with increasing $k^+$ . This breakdown occurs at higher sand-grain roughness Reynolds numbers ( $k_s^+$ ) with increasing solidity. Locally, we find that the heat transfer can be meaningfully partitioned into two categories: exposed, high-shear regions experiencing higher heat transfer obeying a local Reynolds analogy and sheltered, reversed-flow regions experiencing lower and spatially uniform heat transfer. The relative contribution of these distinct mechanisms to the global heat transfer depends on the fraction of the total surface area covered by these regions, which ultimately depends on $\varLambda$ . These insights enable us to develop a model for the rough-wall heat-transfer coefficient, ${C_{h,k}(k^+, \varLambda, Pr)}$ , where $Pr$ is the molecular Prandtl number, that assumes different heat-transfer laws in exposed and sheltered regions. We show that the exposed–sheltered surface-area fractions can be modelled through simple ray tracing that is solely dependent on the surface topography and a prescribed sheltering angle. Model predictions compare well when applied to heat-transfer data of traverse ribs from the literature.

This study investigates the impact of insufficient spanwise spatial resolution on the measurement accuracy of streamwise velocity fluctuations over rough walls. We use a direct numerical simulation (DNS) database of turbulent open-channel flow over three-dimensional sinusoidal roughness with varied wavelengths and roughness heights. Employing a triple decomposition, we investigate both the attenuation of the turbulent fluctuations (about the local mean), u^' and the dispersive stresses (roughness-induced fluctuations of the time-averaged mean about the global mean), Ũ . A boxcar filter on DNS data is applied to investigate the effects of spanwise spatial filtering on these quantities. Our analysis reveals the significance of two key length-scale ratios for velocity measurements over rough walls: the wire length relative to the spatially and temporally plane-averaged Kolmogorov scale at the roughness crest ( l/⟨η⟩ _k ), and the wire length relative to the roughness spanwise wavelength ( l/Λ _y ). We observe that maintaining l/⟨η⟩ _k constant while increasing l/Λ _y attenuates the variance of Ũ and u^' within the roughness sublayer. When fixing l/Λ _y , an increase in l/⟨η⟩ _k influences the turbulent fluctuations across all wall-normal locations. These findings highlight the necessity of considering both length scales when evaluating spanwise spatial resolution in turbulence measurements over rough walls.

The present paper gives an overview of the recent modelling activities under NATO-STO AVT-349, focussed on the understanding and modelling of boundary layers for incompressible, high-Reynolds-number flows subject to non-equilibrium conditions such as strong pressure gradients, three-dimensionality, and surface roughness and heterogeneity. For this, we consider simpler cases where the above flow conditions are present separately or in a reduced number of combinations. First, we focus on the effect of roughness on the outer flow and the problems associated to its characterisation and prediction, with a particular emphasis on the conditions necessary for outer-layer similarity to hold. We then focus on how the presence of adverse and favourable pressure gradients affects the effect of roughness, and to what extent the figures used to quantify it are still useful under such conditions. We also consider the effect of surface heterogeneity, the shortcomings when modelling it and how these can be addressed. We then focus on the effect on the outer layer of pressure gradients and non-equilibrium conditions, to what extent similarity holds in those conditions, and how RANS models perform for such flows, identifying routes for their improvement to handle pressure gradients and non-equilibrium. We also discuss the use of data-driven and machine-aided methods in closure models.

Characterizing the mechanisms that contribute to the onset of drag increase over micro-grooves (riblets) as the spacing increases is critical to design strategies for riblet-based drag reduction. This study decomposes the roughness function to investigate different mechanisms associated with the breakdown of drag reduction as riblet spacing is increased. We obtain the roughness function through direct numerical simulations (DNS) in a minimal channel and restricted nonlinear (RNL) models. Both the traditional RNL decomposition and an augmented RNL (ARNL) model that includes additional nonlinear interactions are employed as computationally tractable, reduced order representations of the flow field. RNL and ARNL results are compared to those of DNS in minimal channels to investigate the role of the different scale-dependent nonlinear interactions contributing to the roughness function. A comparison of the co-spectra arising from the minimal channel DNS with that from RNL and ARNL simulations indicates that general trends are captured by both reduced order models. However, the additional nonlinearity introduced in the ARNL model produces closer correspondence in the observed structural features of the DNS results. In particular, the ARNL better captures the signatures of the dispersive flow and the texture-coherent fluctuations. There is also a noticeable improvement observed in the profiles of the added stress contributions obtained with the ARNL model versus the RNL model.

This paper introduces a viscous vortex model for predicting the optimal drag reduction of riblet surfaces, eliminating the need for expensive direct numerical simulations (DNSs) or experiments. The footprint of a typical quasi-streamwise vortex, in terms of the spanwise and wall-normal velocities, is extracted from smooth-wall DNS flow fields in close proximity to the surface. The extracted velocities are then averaged and used as boundary conditions in a Stokes-flow problem, wherein riblets with various cross-sectional shapes are embedded. Here, the same smooth-wall-based boundary conditions can be used for riblets, as we observe from the DNSs that the quasi-streamwise vortices remain unmodified apart from an offset. In particular, the position of these vortices remain unpinned above small riblets. The present approach is compared with the protrusion-height model of Luchini et al. (J. Fluid Mech., vol. 228, 1991, pp. 87–109), which is also based on a Stokes calculation, but represents the vortex with only a uniform spanwise velocity boundary condition. The key novelty of the present model is the introduction of a wall-normal velocity component into the boundary condition, thus inducing transpiration at the riblet crests, which becomes relevant as the riblet size increases. Consequently, the present model allows for the drag-reduction prediction of riblets up to the optimal size. The present approach does not rely on the scale separation formally required by homogenisation techniques, which are only applicable for vanishingly small riblets.

The effects of roughness were considered as part of a NATO Advanced Vehicle Technology effort titled 'Non-Equilibrium Turbulent Boundary Layers at High Reynolds Numbers' (NATO AVT-349). This paper comments on the current state of understanding of the flow physics and modelling efforts to predict rough-wall boundary layer behaviour. Outer layer similarity to smooth wall flows and Reynolds number effects are discussed for zero, favourable, and adverse pressure gradients based on the results of experiments and numerical simulations. Various types of modelling are considered including Reynolds averaged Navier-Stokes (RANS) models with different roughness and turbulence models, wall-modelled large eddy simulations (WMLES), and resolvent models. Current needs and gaps in present understanding are discussed along with recommendations for future experiments and computations.

Direct numerical simulations of incompressible turbulent forced convection over irregular, anisotropic surface roughness in a pressure-driven plane channel flow have been performed. Heat transfer was simulated by solving the passive scalar transport equation with Prandtl number Pr = 0.7. The roughness topographies under investigation here are based on an X-ray computed tomography scan of an additively manufactured internal cooling passage, which had an irregular, multi-scale and mildly non-Gaussian height distribution. Three different roughness topographies and three different friction Reynolds numbers (Reτ = 395, 590, 720) were considered, along with reference smooth-wall simulations at matched Reτ. By systematically varying the roughness topography and flow conditions, a direct computational assessment of aero-thermal performance (pressure losses and heat transfer) and the Reynolds analogy factor, i.e. 2Ch/Cf, where Ch is the heat-transfer coefficient (Stanton number) and Cf is the skin-friction coefficient, was conducted. The results highlight the profound impact that the roughness orientation (relative to the flow direction) has upon the aero-thermal performance of additively manufactured internal passages, with transverse-aligned roughness augmenting heat transfer by as much as 33%, relative to its streamwise-aligned counterpart. An interrogation of velocity and temperature statistics in the near-wall region was also performed, which underlined the growing dissimilarity between heat transfer and drag as fully-rough conditions are approached.

In the fully rough regime, proposed models predict a scaling for a roughness heat-transfer coefficient, e.g. the roughness Stanton number ${St}_k \sim (k^+)^{-p} {Pr}^{-m}$ where the exponent values $p$ and $m$ are model dependent, giving diverse predictions. Here, $k^+$ is the roughness Reynolds number and ${Pr}$ is the Prandtl number. To clarify this ambiguity, we conduct direct numerical simulations of forced convection over a three-dimensional sinusoidal surface spanning $k^+ = 5.5$ – $111$ for Prandtl numbers ${Pr} = 0.5$ , 1.0 and 2.0. These unprecedented parameter ranges are reached by employing minimal channels, which resolve the roughness sublayer at an affordable cost. We focus on the fully rough phenomenologies, which fall into two groups: $p=1/2$ (Owen & Thomson, J. Fluid Mech. , vol. 15, issue 3, 1963, pp. 321–334; Yaglom & Kader, J. Fluid Mech. , vol. 62, issue 3, 1974, pp. 601–623) and $p=1/4$ (Brutsaert, Water Resour. Res. , vol. 11, issue 4, 1975 b , pp. 543–550). Although we find the mean heat transfer favours the $p=1/4$ scaling, the Prandtl–Blasius boundary-layer ideas associated with the Reynolds–Chilton–Colburn analogy that underpin the $p=1/2$ can remain an apt description of the flow locally in regions exposed to high shear. Sheltered regions, meanwhile, violate this behaviour and are instead dominated by reversed flow, where no clear correlation between heat and momentum transfer is evident. The overall picture of fully rough heat transfer is then not encapsulated by one singular mechanism or phenomenology, but rather an ensemble of different behaviours locally. The implications of the approach to a Reynolds-analogy-like behaviour locally on bulk measures of the Nusselt and Stanton numbers are also examined, with evidence pointing to the onset of a regime transition at even-higher Reynolds numbers.

We use the linearized Navier-Stokes equations to study the large-scale flow structures in unstably stratified turbulent channel flows. The impulse response of the linear operator at bulk Richardson numbers from Ri b = 0 . 001 to Ri b = 1 . 0 are considered, corresponding to the increasing influence of buoyancy relative to shear. We compare the streamwise-constant flow structures predicted by the linear model to the quasistreamwise rolls that emerge with increasing Ri b in direct numerical simulations (DNSs) [e.g., Pirozzoli et al. , J. Fluid Mech. 821 , 482 (2017)]. The linearized Navier-Stokes equations augmented with eddy-viscosity and eddy-diffusivity capture the emergence of the quasistreamwise rolls well. With increasing Ri b , the temperature fluctuations of the streamwise-constant flow structures transition from having a peak in intensity at the channel centerline to having two peaks, one at each wall, consistent with DNS.

Direct numerical simulation is used to study turbulent flow over irregular rough surfaces in the periodic minimal channel configuration. The generation of irregular rough surface is based on a random algorithm, in which the power spectrum of the roughness height function along with its probability density function can be directly prescribed. The hydrodynamic properties of the roughness are investigated and compared to those obtained from full-size DNS for 12 roughness topographies with systematically varied PDF and PS at four roughness height. The comparison confirms the viability of the minimal channel approach for characterization of rough surfaces providing excellent agreement in roughness function and zero-plane displacement across various types of roughness and different regimes. Results also indicates that different realizations of roughness, with a fixed PS and PDF, translate to similar values of roughness function with a small scatter. In addition to the global flow properties, the distribution of time-averaged surface force exerted by the roughness onto the fluid is examined and compared to the roughness height distribution for different cases. It is shown that the surface force distribution has an anisotropic structure with spanwise-elongated coherent regions. The anisotropy translates into a very small streamwise integral length scale, which weakly depends on the considered roughness topography, while the larger spanwise integral length scale shows a stronger dependence on roughness characteristics. It is also shown that the sheltering model describes well the spatial distribution of the surface force. Finally, existing roughness correlations are assessed using the present dataset. It was shown that the most correlations can reproduce the values of equivalent sand-grain roughness from DNS within+-30% error while none of the correlations shows a superior predictive accuracy.

We study the flow above non-optimal riblets, specifically large drag-increasing and two-scale trapezoidal riblets. In order to reach large Reynolds numbers and large scale separation while retaining access to flow details, we employ a combination of boundary-layer hot-wire measurements and direct numerical simulation (DNS) in minimal-span channels. Although the outer Reynolds numbers differ, we observe fair agreement between experiments and DNS at matched viscous–friction-scaled riblet spacings $s^+$ in the overlapping physical and spectral regions, providing confidence that both data sets are valid. We find that hot-wire velocity spectra above very large riblets with $s^+ \gtrsim 60$ are depleted of near-wall energy at scales that are (much) greater than $s$ . Large-scale energy likely bypasses the turbulence cascade and is transferred directly to secondary flows of size $s$ , which we observe to grow in strength with increasing riblet size. Furthermore, the present very large riblets reduce the von Kármán constant $\kappa$ of the spanwise uniform mean velocity in a logarithmic layer and, thus, reduce the accuracy of the roughness-function concept, which we link to the near-wall damping of large flow structures. Half-height riblets in the groove, which we use as a model of imperfectly repeated (spanwise-varying) riblets, impede in-groove turbulence. We show how to scale the drag optimum of imperfectly repeated riblets based on representative measurements of the true geometry by solving inexpensive Poisson equations.

To predict drag on a rough surface under turbulent flow conditions, practitioners rely on roughness correlations that map topographical features of the surface to the equivalent sand-grain roughness k(s). However, details of the data that underpin these empirical correlations are not always immediately evident for comparison and discussion. Therefore, here we compile a table of roughness correlations with unified notation, in chronological order, listing the parameter ranges and the roughness types used in their development, noting idiosyncrasies. Overall, the table shows that tested roughness types have increased in generality from regular roughness features to random surface elevations, and that the independent parameters of primary importance measure size (e.g., height), frontal area (e.g., slope), and coverage (e.g., skewness). In addition to the need for more data to populate the parameter space, outstanding questions facing practitioners are filtering and sampling of multiscale surfaces, and the treatment of heterogeneous surfaces, with answers appearing on the horizon.

In this study, we develop an analytical model to predict the turbulent boundary layer downstream of a step-change in the surface roughness where upstream flow conditions are given. We first revisit the classical model of Elliott (Trans. Am. Geophys. Union, vol. 39, 1958, pp. 1048–1054), who modelled the velocity distribution within and above the internal layer with a simple piecewise logarithmic profile, and evolved the velocity profile using the streamwise momentum equation. Elliott's model was originally developed for an atmospheric surface layer, and to make the model applicable to a spatially developing turbulent boundary layer with finite thickness, we propose a number of more physical refinements, including adding a wake function to the velocity profile, considering the growth of the entire boundary layer in the streamwise direction, and using a more realistic shear stress profile in the momentum equation. In particular, we implement the blending model (Li et al., J. Fluid Mech., vol. 923, 2021, p. A18) to account for the deviation of the mean flow within the internal layer from a canonical velocity profile based on the local wall condition. These refinements lead to improved agreement between the prediction and the measurement, especially in the vicinity of the rough-to-smooth change.