The boundary-layer stability on a section of a rotating wind turbine blade with an FFA-W3 series aerofoil at a chord Reynolds number of $3 \times 10^5$, with varying rotation and radii, is studied with direct numerical simulations and linear stability analyses. Low rotation does not significantly affect transition in the outboard blade region. The relative insensitivity to rotation is due to a laminar separation bubble near the leading edge, spanwise-deformed by a primary self-excited instability, promoting the secondary absolute instability of the Kelvin–Helmholtz (KH) vortices and rapid transition. Moderate increases in rotation, or moving inboard, stabilise the flow by accelerating the attached boundary layer and possibly inducing competition between cross-flow and KH modes. This delays separation and transition. Initially, for high rotation rates or radial locations close to the hub, transition is delayed. Nevertheless, strong stationary and travelling cross-flow modes are eventually triggered, spanwise modulating the KH rolls and shifting the transition line close to the leading edge. Cross-flow velocities as high as $56\,\%$ of the free stream velocity directed towards the blade tip are reached at the transition location. For radial locations farther from the hub, the effective angle of attack is decreased, and cross-flow transition occurs at lower rotation rates. The advance or delay of the transition line compared with a non-rotating configuration depends on the competing rotation effects of stabilising the attached boundary layer and triggering cross-flow modes in the separation flow region.

High-fidelity simulations are conducted to investigate the turbulent boundary layers around a finite-span NACA0012 wing with a rounded wing-tip geometry at a chord-based Reynolds number of $Re_c=200\,000$ and at various angles of attack up to $10^\circ$. The study aims to discern the differences between the boundary layers on the finite-span wing and those on infinite-span wings at equivalent angles of attack. The finite-span boundary layers exhibit: (i) an altered streamwise and a non-zero spanwise pressure gradient as a result of the variable downwash induced by the wing-tip vortices (an inviscid effect typical of finite-span wings); (ii) differences in the flow history at different wall-normal distances, caused by the variable flow angle in the wall-normal direction (due to constant pressure gradients and variable momentum normal to the wall); (iii) laminar flow entrainment into the turbulent boundary layers near the wing tip (due to a laminar–turbulent interface); and (iv) variations in boundary layer thickness across the span, attributed to the variable wall-normal velocity in that direction (a primarily inviscid effect). These physical effects are then used to explain the differences in the Reynolds stress profiles and other boundary layer quantities, including the reduced near-wall peak of the streamwise Reynolds stress and the elevated Reynolds stress levels near the boundary layer edge, both observed in the finite-span wings. Other aspects of the flow, such as the downstream development of wing-tip vortices and their interactions with the surrounding flow, are reserved for future investigations.

Large-scale coherent structures in incompressible turbulent pipe flow are studied for a wide range of Reynolds numbers ($Re_\tau =180, 550, 1000, 2000$ and $5200$). Employing the Karhunen–Loève decomposition and a novel approach based on the Voronoi diagram, we identify and classify statistically coherent structures based on their location, dimensions and $Re_{\tau }$. With increasing $Re_{\tau }$, two distinct classes of structures become more energetic, namely wall-attached and detached eddies. The Voronoi methodology is shown to delineate these two classes without the need for specific criteria or thresholds. At the highest $Re_{\tau }$, the attached eddies scale linearly with the wall-normal distance with a slope of approximately $l_y\sim 1.2y/R$, while the detached eddies remain constant at the size of $l_y \approx 0.26R$, with a progressive shift towards the pipe centre. We extract these two classes of structures and describe their spatial characteristics, including radial size, helix angle and azimuthal self-similarity. The spatial distribution could help explain the differences in mean velocity between pipe and channel flows, as well as in modelling large and very-large-scale motions (LSM and VLSM). In addition, a comprehensive description is provided for both wall-attached and detached structures in terms of LSM and VLSM.

We investigate the global instability mechanism of the flow past a three-dimensional stepped cylinder. A comprehensive study is performed for different diameter ratios of the two joined cylinders ($r=D/d$) ranging from $r=1.1$ to $r=4$. Independently of $r$, the spectrum of the linearised Navier–Stokes operator reveals a pair of complex conjugate eigenvalues, with Strouhal number $St \approx 0.11$. The initial transition is triggered by a two-dimensional mechanism of the larger cylinder only, not affected by the presence of the junction and the smaller cylinder ($Re_{D,cr}\approx 47$). The structural sensitivity analysis is used to identify where the instability mechanism acts. The onset of transition is solely localised in the large cylinder wake (L cell), where the wavemaker has two symmetric lobes across the separation bubble. When the Reynolds number increases, a second and a third unstable pair of complex conjugate eigenvalues appears. They are localised in the small cylinder (S) wake and modulation (N) region. For any $r$, the appearance of unstable eigenmodes resembling the three cells S–N–L in the wake is observed. The nonlinear simulation results support this finding, in contrast with the previous classification of the laminar vortex shedding in direct (L–S) and indirect (L–N–S) modes interaction Lewis & Gharib (Phys. Fluids, vol. 4, 1992, pp. 104–117). This result indicates that each cell undergoes a supercritical Hopf bifurcation for any $r$. As $r$ approaches $1$, the modal linear stability results also show an unstable eigenmode in the wake of the small cylinder resembling a new modulation cell, named N2, similar to the N cell but mirrored with respect to the junction plane.

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$, the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$. Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$. This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.

The turbulent external flow around a three-dimensional stepped cylinder is studied by means of direct numerical simulations with the adaptive mesh refinement technique. We give a broad perspective of the flow regimes from laminar to turbulent wake at $Re_D=5000$, which is the highest ever considered for this flow case. In particular, we focus on the intermediate Reynolds number $Re_D=1000$ that reveals a turbulent wake coupled with a stable cylinder shear layer (subcritical regime). This flow shows a junction dynamics similar to the laminar $Re_D=150$, where no hairpin vortex appears around the edges, and just two horseshoe vortices are visible. A new stable vortex in the form of a ring, which coils around the rear area, is also identified. In the turbulent wake, the presence of three wake cells is pointed out: the large and small cylinder cells together with the modulation region. However, the modulation dynamics varies between the subcritical and turbulent regimes. A time-averaged, three-dimensional set of statistics is computed, and spatially coherent structures are extracted via proper orthogonal decomposition (POD). The POD identifies the (long-debated) connection between the N-cell and the downwash behind the junction. Furthermore, as the Reynolds number increases, the downwash phenomenon becomes less prominent. Eventually, a reduced-order reconstruction with the most energetically relevant modes is defined to explain the wake vortex interactions. This also serves as a valuable starting point for simulating the stepped cylinder wake behaviour within complex frameworks, e.g. fluid–structure interaction.

Multifidelity models (MFMs) can be used to construct predictive models for flow quantities of interest (QoIs) over the space of uncertain/design parameters, with the purpose of uncertainty quantification, data fusion and optimization. For numerical simulation of turbulence, there is a hierarchy of methodologies ranked by accuracy and cost, where each methodology may have several numerical/modelling parameters that control the predictive accuracy and robustness of its resulting outputs. Compatible with these specifications, the present hierarchical MFM strategy allows for simultaneous calibration of the fidelity-specific parameters in a Bayesian framework as developed by Goh et al. (Technometrics, vol. 55, no. 4, 2013, pp. 501–512). The purpose of the MFM is to provide an improved prediction, mainly interpolation over the range covered by training data, by combining lower- and higher-fidelity data in an optimal way for any number of fidelity levels; even providing confidence intervals for the resulting QoI. The capabilities of the MFM are first demonstrated on an illustrative toy problem, and it is then applied to three realistic cases relevant to engineering turbulent flows. The latter include the prediction of friction at different Reynolds numbers in turbulent channel flow, the prediction of aerodynamic coefficients for a range of angles of attack of a standard airfoil and the uncertainty propagation and sensitivity analysis of the separation bubble in the turbulent flow over periodic hills subject to geometrical uncertainties. In all cases, based on only a few high-fidelity data samples, the MFM leads to accurate predictions of the QoIs. The result of the uncertainty quantification and sensitivity analyses are also found to be accurate compared with the ground truth in each case.

The flow physics of turbulent boundary layers is investigated using spectral analysis based on the spanwise scale decomposition of the Reynolds stress transport equation, with data obtained from a direct numerical simulation of the turbulent boundary layer at $Re_\tau \simeq 2020$. Here, we extend the framework of Kawata & Alfredsson (Phys. Rev. Lett., vol. 120, 2018, p. 244501) for plane Couette flows to zero-pressure-gradient boundary layers. The equation contains three fundamental fluxes, which govern the Reynolds stress transport: (i) a scale flux of the interaction between small-scale and large-scale structures, and two spatial fluxes dominated by (ii) pressure and (iii) turbulent transport along the wall-normal direction. The scale flux reveals evidence of the inverse turbulent kinetic energy transfer, from small to large scales, occurring at the near-wall region, whereby for the scale flux of the Reynolds shear stress transport, the inverse transfer extends across the entire boundary layer. The wall-normal fluxes reveal the interactions occurring between scales at the buffer and logarithmic regions. In addition, there is interaction between the large-scale structures and the free stream flow occurring at the edge of the boundary layer, which was not observed in the Couette flow. Flow structures associated with inverse interscale transport of Reynolds shear stress are identified by applying conditional analysis to the spectrally decomposed velocity fields. While the inverse transport is interpreted as the net energy transfer from small-scale ejections ($Q2$) and sweeps ($Q4$) to the large-scale counterparts, conditional time estimates of the direct and inverse interscale transport reveal that both processes play a substantial role across a broad range of scales.

The characteristics of turbulent boundary layers (TBLs) subjected to adverse pressure gradients are analysed through well-resolved large-eddy simulations. The geometries under study are the NACA0012 and NACA4412 wing sections, at $0^{\circ }$ and $5^{\circ }$ angle of attack, respectively, both of them at a Reynolds number based on inflow velocity and chord length of $Re_{c}=400\,000$. The turbulence statistics show that adverse pressure gradients (APGs) have a significant effect on the mean velocity, velocity fluctuations and turbulent kinetic energy budget, and this effect is more prominent on the outer region of the boundary layer. Furthermore, the effect of flow history is assessed by means of an integrated Clauser pressure-gradient parameter $\overline{\unicode[STIX]{x1D6FD}}$ (Vinuesa et al., Flow Turbul. Combust., vol. 99, 2017, pp. 565–587), through the study of cases with matching local values of $\unicode[STIX]{x1D6FD}$ and the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ to isolate this effect. Our results show a noticeable effect of the flow history on the outer region, however the differences in the near-wall peak of the tangential velocity fluctuations appear to be mostly produced by the local APG magnitude. The one-dimensional power-spectral density shows energetic small scales in the outer region of APG TBLs, whereas these energetic scales do not appear in zero-pressure-gradient (ZPG) TBLs, suggesting that small scales near the wall are advected towards the outer layer by the APG. Moreover, the linear coherence spectra show that the spectral outer peak of high-Reynolds-number ZPG TBLs is highly correlated with the near-wall region (Baars et al., J. Fluid Mech., vol. 823, 2017, R2), unlike APG TBLs which do not show such a correlation. This result, together with the different two-dimensional spectra of APG and high-Reynolds-number ZPG TBLs, suggests different energisation mechanisms due to APG and increase in Reynolds number. To the authors’ knowledge, this is the first in-depth analysis of the TBL characteristics over wings, including detailed single-point statistics, spectra and coherence.

Direct numerical simulations (DNSs) are performed to analyse the secondary flow of Prandtl’s second kind in fully developed spanwise-periodic channels with in-plane sinusoidal walls. The secondary flow is characterized for different combinations of wave parameters defining the wall geometry at $Re_{h}=2500$ and 5000, where $h$ is the half-height of the channel. The total cross-flow rate in the channel $Q_{yz}$ is defined along with a theoretical model to predict its behaviour. Interaction between the secondary flows from opposite walls is observed if $\unicode[STIX]{x1D706}\simeq h\simeq A$, where $A$ and $\unicode[STIX]{x1D706}$ are the amplitude and wavelength of the sinusoidal function defining the wall geometry. As the outer-scaled wavelength ($\unicode[STIX]{x1D706}/h$) is reduced, the secondary vortices become smaller and faster, increasing the total cross-flow rate per wall. However, if the inner-scaled wavelength ($\unicode[STIX]{x1D706}^{+}$) is below 130 viscous units, the cross-flow decays for smaller wavelengths. By analysing cases in which the wavelength of the wall is much smaller than the half-height of the channel $\unicode[STIX]{x1D706}\ll h$, we show that the cross-flow distribution depends almost entirely on the separation between the scales of the instantaneous vortices, where the upper and lower bounds are determined by $\unicode[STIX]{x1D706}/h$ and $\unicode[STIX]{x1D706}^{+}$, respectively. Therefore, the distribution of the secondary flow relative to the size of the wave at a given $Re_{h}$ can be replicated at higher $Re_{h}$ by decreasing $\unicode[STIX]{x1D706}/h$ and keeping $\unicode[STIX]{x1D706}^{+}$ constant. The mechanisms that contribute to the mean cross-flow are analysed in terms of the Reynolds stresses and using quadrant analysis to evaluate the probability density function of the bursting events. These events are further classified with respect to the sign of their instantaneous spanwise velocities. Sweeping events and ejections are preferentially located in the valleys and peaks of the wall, respectively. The sweeps direct the instantaneous cross-flow from the core of the channel towards the wall, turning in the wall-tangent direction towards the peaks. The ejections drive the instantaneous cross-flow from the near-wall region towards the core. This preferential behaviour is identified as one of the main contributors to the secondary flow.

This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.

For problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on timestepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing timestepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential timestepping.

This paper introduces a new method based on the diagnostic plot (Alfredsson et al., Phys. Fluids, vol. 23, 2011, 041702) to assess the convergence towards a well-behaved zero-pressure-gradient (ZPG) turbulent boundary layer (TBL). The most popular and well-understood methods to assess the convergence towards a well-behaved state rely on empirical skin-friction curves (requiring accurate skin-friction measurements), shape-factor curves (requiring full velocity profile measurements with an accurate wall position determination) or wake-parameter curves (requiring both of the previous quantities). On the other hand, the proposed diagnostic-plot method only needs measurements of mean and fluctuating velocities in the outer region of the boundary layer at arbitrary wall-normal positions. To test the method, six tripping configurations, including optimal set-ups as well as both under- and overtripped cases, are used to quantify the convergence of ZPG TBLs towards well-behaved conditions in the Reynolds-number range covered by recent high-fidelity direct numerical simulation data up to a Reynolds number based on the momentum thickness and free-stream velocity $Re_{\unicode[STIX]{x1D703}}$ of approximately 4000 (corresponding to 2.5 m from the leading edge) in a wind-tunnel experiment. Additionally, recent high-Reynolds-number data sets have been employed to validate the method. The results show that weak tripping configurations lead to deviations in the mean flow and the velocity fluctuations within the logarithmic region with respect to optimally tripped boundary layers. On the other hand, a strong trip leads to a more energized outer region, manifested in the emergence of an outer peak in the velocity-fluctuation profile and in a more prominent wake region. While established criteria based on skin-friction and shape-factor correlations yield generally equivalent results with the diagnostic-plot method in terms of convergence towards a well-behaved state, the proposed method has the advantage of being a practical surrogate that is a more efficient tool when designing the set-up for TBL experiments, since it diagnoses the state of the boundary layer without the need to perform extensive velocity profile measurements.

Turbulent boundary layers under adverse pressure gradients are studied using well-resolved large-eddy simulations (LES) with the goal of assessing the influence of the streamwise pressure-gradient development. Near-equilibrium boundary layers were characterized through the Clauser pressure-gradient parameter $\unicode[STIX]{x1D6FD}$. In order to fulfil the near-equilibrium conditions, the free stream velocity was prescribed such that it followed a power-law distribution. The turbulence statistics pertaining to cases with a constant value of $\unicode[STIX]{x1D6FD}$ (extending up to approximately 40 boundary-layer thicknesses) were compared with cases with non-constant $\unicode[STIX]{x1D6FD}$ distributions at matched values of $\unicode[STIX]{x1D6FD}$ and friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$. An additional case at matched Reynolds number based on displacement thickness $Re_{\unicode[STIX]{x1D6FF}^{\ast }}$ was also considered. It was noticed that non-constant $\unicode[STIX]{x1D6FD}$ cases appear to approach the conditions of equivalent constant $\unicode[STIX]{x1D6FD}$ cases after long streamwise distances (approximately 7 boundary-layer thicknesses). The relevance of the constant $\unicode[STIX]{x1D6FD}$ cases lies in the fact that they define a ‘canonical’ state of the boundary layer, uniquely characterized by $\unicode[STIX]{x1D6FD}$ and $Re$. The investigations on the flat plate were extended to the flow around a wing section overlapping in terms of $\unicode[STIX]{x1D6FD}$ and $Re$. Comparisons with the flat-plate cases at matched values of $\unicode[STIX]{x1D6FD}$ and $Re$ revealed that the different development history of the turbulent boundary layer on the wing section leads to a less pronounced wake in the mean velocity as well as a weaker second peak in the Reynolds stresses. This is due to the weaker accumulated effect of the $\unicode[STIX]{x1D6FD}$ history. Furthermore, a scaling law suggested by Kitsios et al. (Intl J. Heat Fluid Flow, vol. 61, 2016, pp. 129–136), proposing the edge velocity and the displacement thickness as scaling parameters, was tested on two constant-pressure-gradient parameter cases. The mean velocity and Reynolds-stress profiles were found to be dependent on the downstream development. The present work is the first step towards assessing history effects in adverse-pressure-gradient turbulent boundary layers and highlights the fact that the values of the Clauser pressure-gradient parameter and the Reynolds number are not sufficient to characterize the state of the boundary layer.

A numerical investigation of two locally applied drag-reducing control schemes is carried out in the configuration of a spatially developing turbulent boundary layer (TBL). One control is designed to damp near-wall turbulence and the other induces constant mass flux in the wall-normal direction. Both control schemes yield similar local drag reduction rates within the control region. However, the flow development downstream of the control significantly differs: persistent drag reduction is found for the uniform blowing case, whereas drag increase is found for the turbulence damping case. In order to account for this difference, the formulation of a global drag reduction rate is suggested. It represents the reduction of the streamwise force exerted by the fluid on a plate of finite length. Furthermore, it is shown that the far-downstream development of the TBL after the control region can be described by a single quantity, namely a streamwise shift of the uncontrolled boundary layer, i.e. a changed virtual origin. Based on this result, a simple model is developed that allows the local drag reduction rate to be related to the global one without the need to conduct expensive simulations or measurements far downstream of the control region.

Direct numerical simulations based on the incompressible nonlinear Navier–Stokes equations of the flow over the surface of a rotating disk have been conducted. An impulsive disturbance was introduced and its development as it travelled radially outwards and ultimately transitioned to turbulence has been analysed. Of particular interest was whether the nonlinear stability is related to the linear stability properties. Specifically three disk-edge conditions were considered; (i) a sponge region forcing the flow back to laminar flow, (ii) a disk edge, where the disk was assumed to be infinitely thin and (iii) a physically realistic disk edge of finite thickness. This work expands on the linear simulations presented by Appelquist et al. (J. Fluid. Mech., vol. 765, 2015, pp. 612–631), where, for case (i), this configuration was shown to be globally linearly unstable when the sponge region effectively models the influence of the turbulence on the flow field. In contrast, case (ii) was mentioned there to be linearly globally stable, and here, where nonlinearity is included, it is shown that both cases (ii) and (iii) are nonlinearly globally unstable. The simulations show that the flow can be globally linearly stable if the linear wavepacket has a positive front velocity. However, in the same flow field, a nonlinear global instability can emerge, which is shown to depend on the outer turbulent region generating a linear inward-travelling mode that sustains a transition front within the domain. The results show that the front position does not approach the critical Reynolds number for the local absolute instability, $R=507$. Instead, the front approaches $R=583$ and both the temporal frequency and spatial growth rate correspond to a global mode originating at this position.

The concept of edge states is investigated in the asymptotic suction boundary layer in relation to the receptivity process to noisy perturbations and the nucleation of turbulent spots. Edge tracking is first performed numerically, without imposing any discrete symmetry, in a large computational domain allowing for full spatial localisation of the perturbation velocity. The edge state is a three-dimensional localised structure recurrently characterised by a single low-speed streak that experiences erratic bursts and planar shifts. This recurrent streaky structure is then compared with predecessors of individual spot nucleation events, triggered by non-localised initial noise. The present results suggest a nonlinear picture, rooted in dynamical systems theory, of the nucleation process of turbulent spots in boundary-layer flows, in which the localised edge state plays the role of state-space mediator.

Turbulence in the asymptotic suction boundary layer is investigated numerically at the verge of laminarisation using direct numerical simulation. Following an adiabatic protocol, the Reynolds number $Re$ is decreased in small steps starting from a fully turbulent state until laminarisation is observed. Computations in a large numerical domain allow in principle for the possible coexistence of laminar and turbulent regions. However, contrary to other subcritical shear flows, no laminar–turbulent coexistence is observed, even near the onset of sustained turbulence. High-resolution computations suggest a critical Reynolds number $Re_{g}\approx 270$, below which turbulence collapses, based on observation times of $O(10^{5})$ inertial time units. During the laminarisation process, the turbulent flow fragments into a series of transient streamwise-elongated structures, whose interfaces do not display the characteristic obliqueness of classical laminar–turbulent patterns. The law of the wall, i.e. logarithmic scaling of the velocity profile, is retained down to $Re_{g}$, suggesting a large-scale wall-normal transport absent in internal shear flows close to the onset. In order to test the effect of these large-scale structures on the near-wall region, an artificial volume force is added to damp spanwise and wall-normal fluctuations above $y^{+}=100$, in viscous units. Once the largest eddies have been suppressed by the forcing, and thus turbulence is confined to the near-wall region, oblique laminar–turbulent interfaces do emerge as in other wall-bounded flows, however only transiently. These results suggest that oblique stripes at the onset are a prevalent feature of internal shear flows, but will not occur in canonical boundary layers, including the spatially growing ones.

Direct numerical simulations of the fully developed turbulent flow through a porous square duct are performed to study the effect of the permeable wall on the secondary cross-stream flow. The volume-averaged Navier–Stokes equations are used to describe the flow in the porous phase, a packed bed with porosity ${\it\varepsilon}_{c}=0.95$. The porous square duct is computed at $\mathit{Re}_{b}\simeq 5000$ and compared with the numerical simulations of a turbulent duct with four solid walls. The two boundary layers on the top wall and porous interface merge close to the centre of the duct, as opposed to the channel, because the sidewall boundary layers inhibit the growth of the shear layer over the porous interface. The most relevant feature in the porous duct is the enhanced magnitude of the secondary flow, which exceeds that of a regular duct by a factor of four. This is related to the increased vertical velocity, and the different interaction between the ejections from the sidewalls and the porous medium. We also report a significant decrease in the streamwise turbulence intensity over the porous wall of the duct (which is also observed in a porous channel), and the appearance of short spanwise rollers in the buffer layer, replacing the streaky structures of wall-bounded turbulence. These spanwise rollers most probably result from a Kelvin–Helmholtz type of instability, and their width is limited by the presence of the sidewalls.

Direct numerical simulations of turbulent channel flows at friction Reynolds numbers (Re) of 550, 1000 and 1500 are used to analyse the turbulent production, transfer and dissipation mechanisms in the compound space of scales and wall distances by means of the Kolmogorov equation generalized to inhomogeneous anisotropic flows. Two distinct peaks of scale-energy source are identified. The first, stronger one, belongs to the near-wall cycle. Its location in the space of scales and physical space is found to scale in viscous units, while its intensity grows slowly with $\mathit{Re}$, indicating a near-wall modulation. The second source peak is found further away from the wall in the putative overlap layer, and it is separated from the near-wall source by a layer of significant scale-energy sink. The dynamics of the second outer source appears to be strongly dependent on the Reynolds number. The detailed scale-by-scale analysis of this source highlights well-defined features that are used to make the properties of the outer turbulent source independent of Reynolds number and wall distance by rescaling the problem. Overall, the present results suggest a strong connection of the observed outer scale-energy source with the presence of an outer region of turbulence production whose mechanisms are well separated from the near-wall region and whose statistical features agree with the hypothesis of an overlap layer dominated by attached eddies. Inner–outer interactions between the near-wall and outer source region in terms of scale-energy fluxes are also analysed. It is conjectured that the near-wall modulation of the statistics at increasing Reynolds number can be related to a confinement of the near-wall turbulence production due to the presence of increasingly large production scales in the outer scale-energy source region.