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.

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, 1975b, 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 investigate the Reynolds analogy over riblets, namely the analogy between the fractional increase in Stanton number $C_h$ and the fractional increase in the skin-friction coefficient $C_f$, relative to a smooth surface. We investigate the direct numerical simulation data of Endrikat et al. (Flow Turbul. Combust., vol. 107, 2021, pp. 1–29). The riblet groove shapes are isosceles triangles with tip angles $\alpha = {30}^{\circ }, {60}^{\circ }, {90}^{\circ }$, a trapezoid, a rectangle and a right triangle. The viscous-scaled riblet spacing varies between $s^+ \approx 10$ to $60$. The global Reynolds analogy is primarily influenced by Kelvin–Helmholtz rollers and secondary flows. Kelvin–Helmholtz rollers locally break the Reynolds analogy favourably, i.e. cause a locally larger fractional increase in $C_h$ than in $C_f$. These rollers induce negative wall shear stress patches which have no analogue in wall heat fluxes. Secondary flows at the riblets’ crests are associated with local unfavourable breaking of the Reynolds analogy, i.e. locally larger fractional increase in $C_f$ than in $C_h$. Only the triangular riblets with $\alpha = {30}^{\circ }$ trigger strong Kelvin–Helmholtz rollers without appreciable secondary flows. This riblet shape globally preserves the Reynolds analogy from $s^+ = 21$ to $33$. However, the other riblet shapes have weak or non-existent Kelvin–Helmholtz rollers, yet persistent secondary flows. These riblet shapes behave similarly to rough surfaces. They unfavourably break the global Reynolds analogy, and do so to a greater extent as $s^+$ increases.

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.

A large-scale spanwise and wall-normal array of sonic anemometers in the atmospheric surface layer is used to acquire all three components of instantaneous fluctuating velocity as well as temperature in a range of stability conditions. These data permit investigation of the three-dimensional statistical structure of turbulence structures. Based on a similar dataset, Krug et al. (Boundary-Layer Meteorol., vol. 172, 2019, pp. 199–214) reported a self-similar range of wall-attached turbulence structures under both unstable and near-neutral stability conditions. They considered only a wall-normal array and thus assessed statistical structure in the wall-normal direction, in relation to the streamwise wavelength. The present work extends the view of a self-similar range of turbulence structures, by including the statistical structure in the spanwise direction. Moreover, by analysing the phase shift between synchronized measurements in the spectral domain, it is inferred how a scale-dependent inclination angle in the streamwise/wall-normal plane varies with stability. Results suggest that the self-similar wall-attached structures have similar aspect ratios between streamwise/wall-normal scales and streamwise/spanwise scales such that $\lambda _x/\Delta z : \lambda _x/\Delta y \approx 1$ for both near-neutral and unstable conditions. Under the most unstable conditions, coherent structures with $\lambda _x/\delta = 1$ are inclined at angles as high as $65^{\circ }$ relative to the solid boundary, while larger scales $\lambda _x/\delta = 6$ exhibit inclination angles of approximately $35^{\circ }$. For near-neutral stability conditions, the angle tends towards $12^{\circ }$ for all scales. It is noted that in the near-neutral condition, the structure inclination angle and the aspect ratio – and thus the statistical modelling of coherent structures in the atmospheric surface layer – are highly sensitive to the value of the stability parameter.

The logarithmic dependence of streamwise turbulence intensity has been observed repeatedly in recent experimental and direct numerical simulation data. However, its spectral counterpart, a well-developed $k^{-1}$ spectrum ($k$ is the spatial wavenumber in a wall-parallel direction), has not been convincingly observed from the same data. In the present study, we revisit the spectrum-based attached eddy model of Perry and co-workers, who proposed the emergence of a $k^{-1}$ spectrum in the inviscid limit, for small but finite $z/\delta$ and for finite Reynolds numbers ($z$ is the wall-normal coordinate, and $\delta$ is the outer length scale). In the upper logarithmic layer (or inertial sublayer), a reexamination reveals that the intensity of the spectrum must vary with the wall-normal location at order of $z/\delta$, consistent with the early observation argued with ‘incomplete similarity’. The streamwise turbulence intensity is subsequently calculated, demonstrating that the existence of a well-developed $k^{-1}$ spectrum is not a necessary condition for the approximate logarithmic wall-normal dependence of turbulence intensity – a more general condition is the existence of a premultiplied power-spectral intensity of $O(1)$ for $O(1/\delta ) < k < O(1/z)$. Furthermore, it is shown that the Townsend–Perry constant must be weakly dependent on the Reynolds number. Finally, the analysis is semi-empirically extended to the lower logarithmic layer (or mesolayer), and a near-wall correction for the turbulence intensity is subsequently proposed. All the predictions of the proposed model and the related analyses/assumptions are validated with high-fidelity experimental data (Samie et al., J. Fluid Mech., vol. 851, 2018, pp. 391–415).

This study presents an experimental dataset documenting the evolution of a turbulent boundary layer downstream of a rough-to-smooth surface transition. To investigate the effect of upstream flow conditions, two groups of experiments are conducted. For the Group-Re cases, a nominally constant viscous-scaled equivalent sand grain roughness $k_{s0}^{+}\approx 160$ is maintained on the rough surface, while the friction Reynolds number $Re_{\tau 0}$ ranges from 7100 to 21 000. For the Group-ks cases, $Re_{\tau 0}\approx 14\,000$ is maintained while $k_{s0}^{+}$ ranges from 111 to 228. The wall-shear stress on the downstream smooth surface is measured directly using oil-film interferometry to redress previously reported uncertainties in the skin-friction coefficient recovery trends. In the early development following the roughness transition, the flow in the internal layer is not in equilibrium with the wall-shear stress. This conflicts with the common practise of modelling the mean velocity profile as two log laws below and above the internal layer height, as first proposed by Elliott (Trans. Am. Geophys. Union, vol. 39, 1958, pp. 1048–1054). As a solution to this, the current data are used to model the recovering mean velocity semi-empirically by blending the corresponding rough-wall and smooth-wall profiles. The over-energised large-scale motions leave a strong footprint in the near-wall region of the energy spectrum, the frequency and magnitude of which exhibit dependence on $Re_{\tau 0}$ and $k_{s0}^{+}$, respectively. The energy distribution in near-wall small scales is mostly unaffected by the presence of the outer flow with rough-wall characteristics, which can be used as a surrogate measure to extract the local friction velocity.

The statistical properties of prograde spanwise vortex cores and internal shear layers (ISLs) are evaluated for a series of high-Reynolds-number turbulent boundary layers. The considered flows span a wide range of both Reynolds number and surface roughness. In each case, the largest spanwise vortex cores in the outer layer of the boundary layer have size comparable to the Taylor microscale $\lambda _T$, and the azimuthal velocity of these large vortex cores is governed by the friction velocity ${u_\tau }$. The same scaling parameters describe the average thickness and velocity difference across the ISLs. The results demonstrate the importance of the local large-eddy turnover time in determining the strain rate confining the size of the vortex cores and shear layers. The relevance of the turnover time, and more generally the Taylor microscale, can be explained by a stretching mechanism involving the mutual interaction of coherent velocity structures such as uniform momentum zones with the evolving shear layers separating the structures.

We carry out direct numerical simulations of turbulent flow over riblets, streamwise- aligned grooves that are designed to reduce drag by modifying the near-wall flow. Twenty riblet geometries and sizes are considered, namely symmetric triangular with tip angle $30^\circ$, $60^\circ$ and $90^\circ$, asymmetric triangular, blade and trapezoidal. To save on computational cost, simulations are performed using the minimal-channel flow configuration. With this unprecedented breadth of high-fidelity flow data near the wall, we are able to obtain more general insights into the flow physics of riblets. As observed by García-Mayoral & Jiménez (J. Fluid Mech., vol. 678, 2011, pp. 317–347), we confirm that the drag-change curves of all the present groove geometries better collapse when reported with the viscous-scaled square root of the groove area $\ell _g^+$, rather than the riblet spacing $s^+$. Using a two-dimensional generalization of the Fukagata–Iwamoto–Kasagi identity in difference form we isolate the different drag-change contributions. We show that the drag increase associated with dispersive stresses carried by secondary flows can be as important as the one associated with the turbulent stresses and the pre-eminence of dispersive stresses can be estimated by the groove width at the riblet mean height.

We examine the behaviour of turbulent boundary layers over surfaces composed of spanwise-alternating smooth and rough strips, where the width of the strips $S$ varies such that $0.32\leqslant S/\overline{\unicode[STIX]{x1D6FF}}\leqslant 6.81$, where $\overline{\unicode[STIX]{x1D6FF}}$ is the boundary-layer thickness averaged over one spanwise wavelength of the heterogeneity. The experiments are configured to examine the influences of spanwise variation in wall shear stress over a large $S/\overline{\unicode[STIX]{x1D6FF}}$ range. Hot-wire anemometry and particle image velocimetry (PIV) reveal that the half-wavelength $S/\overline{\unicode[STIX]{x1D6FF}}$ governs the diameter and strength of the resulting mean secondary flows and hence the observed isovels of the mean streamwise velocity. Three possible cases are observed: limiting cases (either $S/\overline{\unicode[STIX]{x1D6FF}}\ll 1$ or $S/\overline{\unicode[STIX]{x1D6FF}}\gg 1$), where the secondary flows are confined near the wall or near the roughness change, and intermediate cases ($S/\overline{\unicode[STIX]{x1D6FF}}\approx 1$), where the secondary flows are space filling and at their strongest. These secondary flows, however, exhibit a time-dependent behaviour which might be masked by time averaging. Further analysis of the energy spectrogram and fluctuating flow fields obtained from PIV show that the secondary flows meander in a similar manner to that of large-scale structures occurring naturally in turbulence over smooth walls. The meandering of the secondary flows is a function of $S/\overline{\unicode[STIX]{x1D6FF}}$ and is most prominent when $S/\overline{\unicode[STIX]{x1D6FF}}\approx 1$.

Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterize surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer.

This paper examines the recovery of the wall-shear stress of a turbulent boundary layer that has undergone a sudden transition from a rough to a smooth surface. Early work of Antonia & Luxton (J. Fluid Mech., vol. 53, 1972, pp. 737–757) questioned the reliability of standard smooth-wall methods for measuring wall-shear stress in such conditions, and subsequent studies show significant disagreement depending on the approach used to determine the wall-shear stress downstream. Here we address this by utilising a collection of experimental databases at $Re_{\unicode[STIX]{x1D70F}}\approx 4100$ that have access to both ‘direct’ and ‘indirect’ measures of the wall-shear stress to understand the recovery to equilibrium conditions of the new surface. Our results reveal that the viscous region ($z^{+}\lesssim 4$) recovers almost immediately to an equilibrium state with the new wall conditions; however, the buffer region and beyond takes several boundary layer thicknesses before recovering to equilibrium conditions, which is longer than previously thought. A unique direct numerical simulation database of a wall-bounded flow with a rough-to-smooth wall transition is employed to confirm these findings. In doing so, we present evidence that any estimate of the wall-shear stress from the mean velocity profile in the buffer region or further away from the wall tends to underestimate its magnitude in the near vicinity of the rough-to-smooth transition, and this is likely to be partly responsible for the large scatter of recovery lengths to equilibrium conditions reported in the literature. Our results also reveal that smaller energetic scales in the near-wall region recover to an equilibrium state associated with the new wall conditions within one boundary layer thickness downstream of the transition, while larger energetic scales exhibit an over-energised state for several boundary layer thicknesses downstream of the transition. Based on these observations, an alternative approach to estimating the wall-shear stress from the premultiplied energy spectrum is proposed.

Direct numerical simulations (DNS) are reported for open-channel flow over streamwise-alternating patches of smooth and fully rough walls. The rough patch is a three-dimensional sinusoidal surface. Owing to the streamwise periodicity, the flow configuration consists of a step change from smooth to rough, and a step change from rough to smooth. The friction Reynolds number varies from 437 over the smooth patch to 704 over the rough patch. Through the fully resolved DNS dataset it is possible to explore many detailed aspects of this flow. Two aspects motivate this work. The first one is the equilibrium assumption that has been widely used in both experiments and computations. However, it is not clear where this assumption is valid. The detailed DNS data reveal a significant departure from equilibrium, in particular over the smooth patch. Over this patch, the mean velocity is recovered up to the beginning of the log layer after a fetch of five times the channel height. However, over the rough patch, the same recovery level is reached after a fetch of two times the channel height. This conclusion is arrived at by assuming that an error of up to 5 % is acceptable and the log layer, classically, starts from 30 wall units above the wall. The second aspect is the reported internal boundary-layer (IBL) growth rates in the literature, which are inconsistent with each other. This is conjectured to be partly caused by the diverse IBL definitions. Five common definitions are applied for the same DNS dataset. The resulting IBL thicknesses are different by 100 %, and their apparent power-law exponents are different by 50 %. The IBL concept, as a layer within which the flow feels the surface underneath, is taken as the basis to search for the proper definition. The definition based on the logarithmic slope of the velocity profile, as proposed by Elliot (Trans. Am. Geophys. Union, vol. 39, 1958, pp. 1048–1054), yields better consistency with this concept based on turbulence characteristics.

This paper quantifies the instantaneous form of large-scale turbulent structures in canonical smooth-wall boundary layers, demonstrating that they adhere to a form that is consistent with the self-sustaining streak instability model suggested by Flores & Jiménez (Phys. Fluids, vol. 22, 2010, 071704) and Hwang & Cossu (Phys. Fluids, vol. 23, 2011, 061702). Our motivation for this study stems from previous observations of large-scale streaks that have been spatially locked in position within spanwise-heterogeneous boundary layers. Here, using similar tools, we demonstrate that the randomly occurring large-scale structures in canonical layers show similar behaviour. Statistically, we show that the signature of large-scale coherent structures exhibits increasing meandering behaviour with distance from the wall. At the upper edge of the boundary layer, where these structures are severely misaligned from the main-flow direction, the induced velocities associated with the strongly yawed vortex packets/clusters yield a significant spanwise-velocity component leading to an apparent oblique coherence of spanwise-velocity fluctuations. This pronounced meandering behaviour also gives rise to a dominant streamwise periodicity at a wavelength of approximately $6\unicode[STIX]{x1D6FF}$. We further statistically show that the quasi-streamwise roll-modes formed adjacent to these very large wavy motions are often one-sided (spanwise asymmetric), in stark contrast to the counter-rotating form suggested by conventional conditionally averaged representations. To summarise, we sketch a representative picture of the typical large-scale structures based on the evidence gathered in this study.

We investigate the behaviour of large-scale coherent structures in a spanwise-heterogeneous turbulent boundary layer, using particle image velocimetry on multiple orthogonal planes. The statistical three-dimensionality is imposed by a herringbone riblet surface, although the key results presented here will be common to many cases of wall turbulence with embedded secondary flows in the form of mean streamwise vortices. Instantaneous velocity fields in the logarithmic layer reveal elongated low-momentum streaks located over the upwash-flow region, where their spanwise spacing is forced by the $2\unicode[STIX]{x1D6FF}$ periodicity of the herringbone pattern. These streaks largely resemble the turbulence structures that occur naturally (and randomly located) in spanwise-homogeneous smooth-/rough-wall boundary layers, although here they are directly formed by the roughness pattern. In the far outer region, the large spanwise spacing permits the streaks to aggressively meander. The mean secondary flows are the time-averaged artefact of the unsteady and spanwise asymmetric large-scale roll modes that accompany these meandering streaks. Interestingly, this meandering, or instability, gives rise to a pronounced streamwise periodicity (i.e. an alternating coherent pattern) in the spatial statistics, at wavelengths of approximately 4.5$\unicode[STIX]{x1D6FF}$. Overall, the observed behaviours largely resemble the streak-instability model that has been proposed for the buffer region, only here at a much larger scale and at a forced spanwise spacing. This observation further confirms recent observations that such features may occur at an entire hierarchy of scales throughout the turbulent boundary layer.

Synthetic jets are zero-net-mass-flux actuators that can be used in a range of flow control applications. For some applications, the scaling of the trajectory of the jet with actuation and cross-flow parameters is important. This scaling is investigated for changes in the friction Reynolds number, changes in the velocity ratio (defined as the ratio between the mean jet blowing velocity and the free-stream velocity) and changes in the actuation frequency of the jet. A distinctive aspect of this study is the high-Reynolds-number turbulent boundary layers (up to $Re_{\unicode[STIX]{x1D70F}}=12\,800$) of the cross-flow. To our knowledge, this is the first study to investigate the effect of the friction Reynolds number of the cross-flow on the trajectory of an (unsteady) jet, as well as the first study to systematically investigate the scaling of the trajectory with actuation frequency. A broad range of parameters is varied (rather than an in-depth investigation of a single parameter) and the results of this study are meant to indicate the relative importance of each parameter rather than the exact influence on the trajectory. Within the range of parameters explored, the critical ones are found to be the velocity ratio as well as a non-dimensional frequency based on the jet actuation frequency, the cross-flow velocity and the jet dimensions. The Reynolds number of the boundary layer is shown to have only a small effect on the trajectory. An expression for the trajectory of the jet is derived from the data, which (in the limit) is consistent with known expressions for the trajectory of a steady jet in a cross-flow.

The spatial signature of spanwise velocity coherence in turbulent boundary layers has been studied using a series of unique large-field-of-view multicamera particle image velocimetry experiments, which were configured to capture streamwise/spanwise slices of the boundary layer in both the logarithmic and the wake regions. The friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}\approx 2600$ was chosen to nominally match the simulation of Sillero et al. (Phys. Fluids, vol. 26 (10), 2014, 105109), who had previously reported oblique features of the spanwise coherence at the top edge of the boundary layer based on the sign of the spanwise velocity, and here we find consistent observations from experiments. In this work, we show that these oblique features in the spanwise coherence relate to the intermittent turbulent bulges at the edge of the layer, and thus the geometry of the turbulent/non-turbulent interface, with the clear appearance of two counter-oriented oblique features. Further, these features are shown to be also present in the logarithmic region once the velocity fields are deconstructed based on the sign of both the spanwise and the streamwise velocity, suggesting that the often-reported meandering of the streamwise-velocity coherence in the logarithmic region is associated with a more obvious diagonal pattern in the spanwise velocity coherence. Moreover, even though a purely visual inspection of the obliqueness in the spanwise coherence may suggest that it extends over a very large spatial extent (beyond many boundary layer thicknesses), through a conditional analysis, we show that this coherence is limited to distances nominally less than two boundary layer thicknesses. Interpretation of these findings is aided by employing synthetic velocity fields of a boundary layer constructed using the attached eddy model, where the range of eddy sizes can be prescribed. Comparisons between the model, which employs an array of self-similar packet-like eddies that are randomly distributed over the plane of the wall, and the experimental velocity fields reveal a good degree of agreement, with both exhibiting oblique features in the spanwise coherence over comparable spatial extents. These findings suggest that the oblique features in the spanwise coherence are likely to be associated with similar structures to those used in the model, providing one possible underpinning structural composition that leads to this behaviour. Further, these features appear to be limited in spatial extent to only the order of the large-scale motions in the flow.

An assessment of the turbulent boundary layer flow structure, which is coherent with the near-wall region, is carried out through a spectral coherence analysis. This spectral method is applied to datasets comprising synchronized two-point streamwise velocity signals at a near-wall reference position and a range of wall-normal positions spanning a Reynolds-number range $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3}){-}O(10^{6})$. Within each dataset, a self-similar structure is identified from the coherence between the turbulence in the logarithmic region and at the near-wall reference position. This self-similarity is described by a streamwise/wall-normal aspect ratio of $\unicode[STIX]{x1D706}_{x}/z\approx 14$, where $\unicode[STIX]{x1D706}_{x}$ and $z$ are the streamwise wavelength and wall-normal distance respectively.

In this paper we examine the characteristics of the interfaces that demarcate regions of relatively uniform streamwise momentum in turbulent boundary layers. The analysis utilises particle image velocimetry databases that span more than an order of magnitude of friction Reynolds number ($Re_{\unicode[STIX]{x1D70F}}=10^{3}$–$10^{4}$), enabling us to provide a detailed description of the interfacial layers as a function of Reynolds number. As reported by Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54), these interfaces appear as persistent regions of strong shear with distinct patches of vorticity consistent with a packet-like structure. Here, however, we treat these interfaces as continuous lines, thus averaging the properties of the vortical patches, and find that their geometry is highly contorted and exhibits self-similarity across a wide range of scales. Specifically, the lengths of the edges of uniform momentum zones exhibit a power-law behaviour with a fractal scaling that has a constant exponent across the boundary layer, while the topmost edge or the turbulent/non-turbulent interface shows a sudden increase in the exponent. The accompanying sharp changes in velocity that occur at these edges are found to change in magnitude as a function of wall-normal height, being larger closer to the wall. Further, a Reynolds number invariance is exhibited when the magnitude of the step-like changes in velocity is scaled by the skin-friction velocity, meanwhile, the width across which it occurs is shown to be of the order of the Taylor microscale. Based on these quantitative measures, the Reynolds number scaling observed and the persistent presence of sharp changes in momentum in turbulent boundary layers, a simple model is used to reconstruct the mean velocity profile. Insight gained from the model enhances our understanding of how instantaneous phenomena (such as a zonal-like structural arrangement) manifests in the averaged flow statistics and confirms that the instantaneous momentum in a turbulent boundary layer appears to mainly consist of a step-like profile as a function of wall-normal distance.

Structural properties of regions of uniform streamwise momentum in turbulent boundary layers are examined using experimental databases obtained from particle image velocimetry. This investigation employs a large range of Reynolds numbers, spanning more than an order of magnitude ($Re_{{\it\tau}}=10^{3}{-}10^{4}$), enabling us to provide a detailed description of uniform momentum zones as a function of Reynolds number. Our analysis starts by examining the identification criterion used by Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) to report the presence of uniform momentum zones in turbulent boundary layers. This criterion is then applied to show that a zonal-like structural arrangement is prevalent in all datasets examined, emphasising its importance in the structural organisation. Streamwise velocity fluctuations within the zones are observed to be small but they are bounded by distinct step changes in streamwise momentum which indicate that shear layers of intense vorticity separate each zone. A log-linear increase in the number of these zones with increasing Reynolds number is revealed, together with an increase in the thicknesses of zones with increasing distance from the wall. These results support a hierarchical length-scale distribution of coherent structures, which generate zonal-like organisation within turbulent boundary layers. Interpretation of these findings is aided by employing synthetic velocity fields generated using a model based on the attached eddy hypothesis, which is described in the work of Perry and co-workers. Comparisons between the model and experimental results show that a hierarchy of self-similar structures leads to population densities and length-scale distributions of uniform momentum zones that closely adhere to those observed experimentally in this study.