Boundary layers are present in many natural and industrial fluid flows. The concept of boundary layers can be traced back to Leonardo da Vinci's paintings of pipe flow, where he was aware of a higher velocity away from the walls. During the 19th century, the physics of boundary conditions had been extensively debated, and the well-known Maxwell–Navier slip length was proposed in 1823. In most cases, the no-slip boundary condition is valid at a fluid–solid interface. However, with the advancement of measurement techniques, slip lengths ranging from nanometre to micrometre scales were experimentally measured, raising questions regarding the applicability of the no-slip condition. In 2003, Lauga & Stone (J. Fluid Mech., vol. 489, 2003, pp. 55–77) proposed a simple model to elucidate the effect of surface heterogeneities on the slip length, elegantly bridging the microscopic structure of the wall-boundary conditions to the macroscopic effective slip length.

Superhydrophobic (SHPo) surfaces can capture a thin layer of air called a plastron under water to reduce skin friction. Although a ~30 % drag reduction has been recently reported with longitudinal micro-trench SHPo surfaces under a boat and in a towing tank, the results lacked the consistency to establish a clear trend. Designed based on Yu et al. (J. Fluid Mech, vol. 962, 2023, A9), this work develops and tests a series of high-performance SHPo surface coupons that can sustain a pinned plastron underneath a passenger motorboat revamped to reach 14 knots. Importantly, plastrons in a pinned state, not just their existence, are confirmed during flow experiments for the first time. All the drag-reduction data measured on different longitudinal micro-trenches are found to collapse if plotted against slip length in wall units. In comparison, aligned posts and transverse trenches show less and little drag reduction, respectively, confirming the adverse effect of the spanwise slip in turbulent flows. This report not only verifies SHPo surfaces can provide a consistent drag reduction at high speeds in open sea but also shows that one may predict the amount of drag reduction in turbulent flows using the physical slip length obtained for Stokes flows.

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

The present study investigates streamwise ($\overline {u^2}$) energy-transfer mechanisms in the inner and outer regions of turbulent boundary layers (TBLs). Particular focus is placed on the $\overline {u^2}$ production, its inter-component and wall-normal transport as well as dissipation, all of which become statistically significant in the outer region with increasing friction Reynolds number ($Re_{\tau }$). These properties are analysed using published data sets of zero, weak and moderately strong adverse-pressure-gradient (APG) TBLs across a decade of $Re_{\tau }$, revealing similarity in energy-transfer pathways for all these TBLs. It is found that both the inner and outer peaks of $\overline {u^2}$ are always associated with local maxima in the $\overline {u^2}$ production and its inter-component transport, and the regions below/above each of these peaks are always dominated by wall-ward/away-from-wall transport of $\overline {u^2}$, thereby classifying the $\overline {u^2}$ profiles into four distinct regimes. This classification reveals existence of phenomenologically similar energy-transfer mechanisms in the ‘inner’ and ‘outer’ regions of moderately strong APG TBLs, which meet at an intermediate location coinciding with the minimum in $\overline {u^2}$ profiles. Conditional averaging suggests existence of similar phenomena even in low $Re_{\tau }$ canonical and/or weak APG TBLs, albeit with the outer-region mechanisms weaker than those in the inner region. This explains the absence of their $\overline {u^2}$ outer peak and the dominance of $\overline {u^2}$ wall-normal transport away from the wall, which potentially originates from the inner region. Given that the wall-ward/away-from-wall transport of $\overline {u^2}$ is governed by the $Q_4$(sweeps)/$Q_2$(ejections) quadrants of the Reynolds shear stress, it is argued that the emergence of the $\overline {u^2}$ outer peak corresponds with the statistical dominance of $Q_4$ events in the outer region. Besides unravelling the dynamical significance of $Q_2$ and $Q_4$ events in the outer region of TBLs, the present analysis also proposes new phenomenological arguments for testing on canonical wall-turbulence data at very high $Re_{\tau }$.

High-frequency observation data, including all three components of instantaneous fluctuating velocity, temperature, as well as particulate matter 10 ($PM_{10}$), collected from the unstable atmospheric surface layer at $z/L = -0.11$ and $-$0.12, $L$ being the Obukhov length, during sand and dust storms (SDS), were used to explore the scaling of vertical coherence and the logarithmic energy profile for wall-attached eddies. The present results demonstrate good agreement with the self-similar range of the wall-attached features for velocity and temperature components, as well as for $PM_{10}$ at lower heights ($z<15$ m) during SDS. Following the idea depicted by Davenport (Q. J. R. Meteorol., vol. 372, 1961, pp. 194–211), an empirically derived transfer kernel comprises implicit filtering via a scale-dependent gain and phase, parametrically defined as $|H_L^2(f)|=\exp (c_1-c_2\delta /\lambda _x)$, where $c_1$ and $c_2$ are parameters, $\delta$ is the boundary layer thickness and $\lambda _x$ is the streamwise wavelength. Linear coherence spectrum analysis is applied as a filter to separate the coherent and incoherent portions. After this separation procedure, the turbulence intensity decay for wall-attached eddies is described in a log–linear manner, which also identifies how the scaling parameter differs between the measured components. These findings present abundant features of wall-attached eddies during SDS which further are used to improve/enrich existing near-wall models.

In this experimental study, we explore the dynamics of the thermal boundary layer in liquid metal Rayleigh–Bénard convection, covering the parameter ranges of $0.026 \leq$ Prandtl numbers $(Pr) \leq 0.033$ and Rayleigh numbers ($Ra$) up to $2.9\times 10^9$. Our research focuses on characterising the thermal boundary layer near the top plate of a cylindrical convection cell with an aspect ratio of 0.5, distinguishing between two distinct regions: the shear-dominated region around the centre of the top plate and a location near the side wall where the boundary layer is expected to be affected by the impact or ejection of thermal plumes. The dependencies of the boundary layer thickness on $Ra$ at these positions reveal deviating scaling exponents with the difference diminishing as $Ra$ increases. We find stronger fluctuations in the boundary layer and increasing deviation from the Prandtl–Blasius–Pohlhausen profile with increasing $Ra$, as well as in the measurements outside the centre region. Our data illustrate the complex interplay between flow dynamics and thermal transport in low-$Pr$ convection.

The influence of symmetry-breaking effects of ridge-type roughness on secondary currents in turbulent channel flow is investigated using direct numerical simulations. The ridges have triangular cross-section, which is systematically varied from isosceles to right-angled triangle, introducing an imbalance to the slopes of the ridges’ lateral surfaces while the streamwise homogeneity of the surfaces is maintained. In all cases, secondary current vortices are produced, but asymmetric ridge cross-sections break the symmetry of these vortices. As a result of the asymmetry-induced misalignment and imbalance in the secondary current vortices, net spanwise flow emerges. The magnitude of the spanwise flow increases with the slope ratio of the ridge lateral surfaces and significantly modifies the mean flow topology, leading to the merging of critical points in the case of the right-angled triangular ridge shape. Within the cavities, the net spanwise flow is accompanied by a non-zero mean spanwise pressure gradient, while from the perspective of the outer flow, the scalene ridge surfaces have a similar effect as a wall that is slowly moving in the spanwise direction. Overall, the present results suggest the existence of a special type of Prandtl's secondary currents of the second kind, namely those that result in net spanwise flow.

This study proposes a novel super-resolution (or SR) framework for generating high-resolution turbulent boundary layer (TBL) flow from low-resolution inputs. The framework combines a super-resolution generative adversarial neural network (SRGAN) with down-sampling modules (DMs), integrating the residual of the continuity equation into the loss function. The DMs selectively filter out components with excessive energy dissipation in low-resolution fields prior to the super-resolution process. The framework iteratively applies the SRGAN and DM procedure to fully capture the energy cascade of multi-scale flow structures, collectively termed the SRGAN-based energy cascade reconstruction framework (EC-SRGAN). Despite being trained solely on turbulent channel flow data (via ‘zero-shot transfer’), EC-SRGAN exhibits remarkable generalization in predicting TBL small-scale velocity fields, accurately reproducing wavenumber spectra compared to direct numerical simulation (DNS) results. Furthermore, a super-resolution core is trained at a specific super-resolution ratio. By leveraging this pretrained super-resolution core, EC-SRGAN efficiently reconstructs TBL fields at multiple super-resolution ratios from various levels of low-resolution inputs, showcasing strong flexibility. By learning turbulent scale invariance, EC-SRGAN demonstrates robustness across different TBL datasets. These results underscore the potential of EC-SRGAN for generating and predicting wall turbulence with high flexibility, offering promising applications in addressing diverse TBL-related challenges.

The scale-dependent variability of convective velocities and structure inclination angles in wall-bounded turbulence was studied experimentally via space–time energy spectrum measurements. We found that the variability of convection velocities for large-scale motions (LSMs) decreased inversely with streamwise wavenumbers, and that the variability trend was not fully explained by earlier applications of Kraichnan's ‘random-sweeping’ model of turbulence that assumed perfect scale separation. By analytically extending the random-sweeping model to allow for a dominant large scale in the random-sweeping signal that can interact with other LSMs, we showed how scale interactions can explain the variability trend in convection velocities for LSMs. The variability in convection velocities was also shown to correlate with the scale-dependent inclination angles of coherent structures that were obtained via cross-spectral analysis. Large-scale motions tended to exhibit shallower inclination to the wall with increasing convection velocity, while small-scale motions and those far from the wall exhibited the reverse behaviour. We proposed that these two opposite relationships between inclination angle and convection velocity can be explained in terms of a balance between opposing effects of the mean shear and the coherent structure geometry. Descriptions and models of convection velocity variability effects are useful both for modelling turbulence spectra and explaining the geometry of coherent structures.

A new wall-wake law is proposed for the streamwise turbulence in the outer region of a turbulent boundary layer. The formulation pairs the logarithmic part of the profile (with a slope $A_1$ and additive constant $B_1$) to an outer linear part, and it accurately describes over 95 % of the boundary layer profile at high Reynolds numbers. Once the slope $A_1$ is fixed, $B_1$ is the only free parameter determining the fit. Most importantly, $B_1$ is shown to follow the same trend with Reynolds number as the wake factor in the wall-wake law for the mean velocity, which is tied to changes in scaling of the mean flow and the turbulence that occur at low Reynolds number.

Two sets of measurements utilizing hot-wire anemometry and oil-film interferometry for flat-plate turbulent boundary layers, exposed to various controlled adverse and favourable pressure gradients, are used to evaluate history effects of the imposed and varying free-stream gradients. The results are from the NDF wind tunnel at Illinois Tech (IIT) and the MTL wind tunnel at KTH, over the range $800 < Re_\tau < 22\,000$ (where $Re_{\tau }$ is the friction Reynolds number). The streamwise pressure-gradient parameter $\beta \equiv (-\ell /\tau _{w}) \cdot (\partial P_{e}/\partial x)$ varied between $-2 < \beta < 7$, where $\ell$ is an outer length scale for boundary layers equivalent to the half-height of channel flow and the radius of pipe flow, and is estimated for each boundary-layer profile; note that $\tau_w$ is the wall-shear stress and $P_e$ is the free-stream static pressure. Extracting from each profile the three parameters of the overlap region, following the recent work of Monkewitz & Nagib (J. Fluid Mech., vol. 967, 2023, p. A15) that led to an overlap region of combined logarithmic and linear parts, we find minimum history effects in the overlap region. Thus, the overlap region in this range of pressure-gradient boundary layers appears to be in ‘quasiequilibrium’.

The topology of turbulent coherent structures is known to regulate the transport of energy, mass and momentum in the atmospheric boundary layer (ABL). While previous research has primarily focused on characterizing the structure of turbulence in stationary ABL flows, real-world scenarios frequently deviate from stationarity, giving rise to nuanced and poorly understood changes in the turbulence geometry and associated transport mechanisms. This study sheds light on this problem by examining topological changes in ABL turbulence induced by non-stationarity and their effects on momentum transport. Results from a large-eddy simulation of pulsatile open channel flow over an array of surface-mounted cuboids are examined. The analysis reveals that the flow pulsation triggers a phase-dependent shear rate, and the ejection-sweep pattern varies with the shear rate during the pulsatile cycle. From a turbulence structure perspective, it is attributed to the changes in the geometry of hairpin vortices. An increase (decrease) in the shear rate intensifies (relaxes) these structures, leading to an increase (decrease) in the frequency of ejections and an amplification (reduction) of their percentage contribution to the total momentum flux. Furthermore, the size of the hairpin packets undergoes variations, which depend on the geometry of the constituting hairpin vortices, yet the packet inclination preserves its orientation throughout the pulsatile cycle. These observations reinforce the important role non-stationarity holds in shaping the structure of ABL turbulence and the momentum transport mechanisms it governs.

We introduce a numerical strategy to study the evolution of two-dimensional water waves in the presence of a plunging jet. The free-surface Navier–Stokes solution is obtained with a finite, but small, viscosity. We observe the formation of a surface boundary layer where the vorticity is localised. We highlight convergence to the inviscid solution. The effects of dissipation on the development of a singularity at the tip of the wave is also investigated by characterising the vorticity boundary layer appearing near the interface.

The impact of bulk viscosity is unclear with considering the increased dilatational dissipation and compressibility effects in hypersonic turbulence flows. In this study, we employ direct numerical simulations to conduct comprehensive analysis of the effect of bulk viscosity on hypersonic turbulent boundary layer flow over a flat plate. The results demonstrate that the scaling relations remain valid even when accounting for large bulk viscosity. However, the wall-normal velocity fluctuations $v_{rms}^{\prime \prime }$ decrease significantly in the viscous sublayer due to the enhanced bulk dilatational dissipation. The intensity of travelling-wave-like alternating positive and negative structures of instantaneous pressure fluctuations $p_{rms}^{\prime }$ in the near-wall region decreases distinctly after considering the bulk viscosity, which is attributed mainly to the reduction of compressible pressure fluctuations $p_{c,rms}^{+}$. Furthermore, the velocity divergence $\partial u_{i} / \partial x_{i}$ undergoes a significant decrease by bulk viscosity. In short, our results indicate that bulk viscosity can weaken the compressibility of the hypersonic turbulent boundary layer and becomes more significant as the Mach number increases and the wall temperature decreases. Notably, when the bulk-to-shear viscosity ratio of the gas reaches a few hundred levels ($\mu _b/\mu =O(10^2)$), and mechanical behaviour of the near-wall region ($\kern 0.06em y^+\le 30$) is of greater interest, the impact of bulk viscosity on the hypersonic cold-wall turbulent boundary layer may not be negligible.

The numerical investigation focuses on the flow patterns around a rectangular cylinder with three aspect ratios ($L/D=5$, $10$, $15$) at a Reynolds number of $1000$. The study delves into the dynamics of vortices, their associated frequencies, the evolution of the boundary layer and the decay of the wake. Kelvin–Helmholtz (KH) vortices originate from the leading edge (LE) shear layer and transform into hairpin vortices. Specifically, at $L/D=5$, three KH vortices merge into a single LE vortex. However, at $L/D=10$ and $15$, two KH vortices combine to form a LE vortex, with the rapid formation of hairpin vortex packets. A fractional harmonic arises due to feedback from the split LE shear layer moving upstream, triggering interaction with the reverse flow. Trailing edge (TE) vortices shed, creating a Kármán-like street in the wake. The intensity of wake oscillation at $L/D=5$ surpasses that in the other two cases. Boundary layer transition occurs after the saturation of disturbance energy for $L/D=10$ and $15$, but not for $L/D=5$. The low-frequency disturbances are selected to generate streaks inside the boundary layer. The TE vortex shedding induces the formation of a favourable pressure gradient, accelerating the flow and fostering boundary layer relaminarization. The self-similarity of the velocity defect is observed in all three wakes, accompanied by the decay of disturbance energy. Importantly, the decrease in the shedding frequency of LE (TE) vortices significantly contributes to the overall decay of disturbance energy. This comprehensive exploration provides insights into complex flow phenomena and their underlying dynamics.

In two-fluid simulations of gas–solid fluidised beds, the gaseous phase and the particulate phase are modelled as continuous media. The stress exerted by the particulate medium on the container walls should be modelled to predict accurately the bed dynamics. This paper addresses the modelling of sliding particle–wall contacts in two-fluid simulations, based on reference simulations coupling computational fluid dynamics with the discrete element method (CFD-DEM), in which the individual movement of the particles is tracked. The analysis of the CFD-DEM highlights the complex near-wall behaviour of the particles, which is not reproduced by two-fluid models. Nevertheless, the particle–wall shear stress can be expressed based on the total granular pressure within the first cell off the wall. The model is validated for the two-fluid simulation of a bubbling gas–solid fluidised bed of olefin particles in the dense-fluidisation regime.

In the context of large off-shore wind farms, power production is influenced greatly by the turbine array's interaction with the atmospheric boundary layer. One of the most influencing manifestations of such complex interaction is the increased level of shear stress observed within the farm. This leads to higher momentum fluxes that affect the wind speed at the turbine locations and in the cluster wake. At the wind farm entrance, an internal boundary layer (IBL) grows due to the change in effective roughness imposed by the wind turbines, and for large enough clusters, this can reach the unperturbed boundary layer height in what is referred to as the fully developed regime. Downwind, a second IBL starts growing, while the shear stress profile decays exponentially to its unperturbed state. In the present study, we propose a simple analytical model for the vertical profile of the horizontal shear stress components in the three regions identified above. The model builds upon the top-down model of Meneveau (J. Turbul., vol. 13, 2012, N7), and assumes that the flow develops in a conventionally neutral boundary layer. The proposed parametrization is verified successfully against large-eddy simulations, demonstrating its ability to capture the vertical profile of horizontal shear stress, and its evolution both inside and downwind of the wind farm. Our findings suggest that the developed model can prove extremely useful to enhance the physical grounds on which new classes of coupled wind farm engineering models are based, leading to a better estimation of meso-scale phenomena affecting the power production of large turbine arrays.

Wall-pressure fluctuations are a practically robust input for real-time control systems aimed at modifying wall-bounded turbulence. The scaling behaviour of the wall-pressure–velocity coupling requires investigation to properly design a controller with such input data so that it can actuate upon the desired turbulent structures. A comprehensive database from direct numerical simulations (DNS) of turbulent channel flow is used for this purpose, spanning a Reynolds-number range $Re_\tau \approx 550\unicode{x2013}5200$. Spectral analysis reveals that the streamwise velocity is most strongly coupled to the linear term of the wall pressure, at a Reynolds-number invariant distance-from-the-wall scaling of $\lambda _x/y \approx 14$ (and $\lambda _x/y \approx 8$ for the wall-normal velocity). When extending the analysis to both homogeneous directions in $x$ and $y$, the peak coherence is centred at $\lambda _x/\lambda _z \approx 2$ and $\lambda _x/\lambda _z \approx 1$ for $p_w$ and $u$, and $p_w$ and $v$, respectively. A stronger coherence is retrieved when the quadratic term of the wall pressure is concerned, but there is only little evidence for a wall-attached-eddy type of scaling. An experimental dataset comprising simultaneous measurements of wall pressure and velocity complements the DNS-based findings at one value of $Re_\tau \approx 2$k, with ample evidence that the DNS-inferred correlations can be replicated with experimental pressure data subject to significant levels of (acoustic) facility noise. It is furthermore shown that velocity-state estimations can be achieved with good accuracy by including both the linear and quadratic terms of the wall pressure. An accuracy of up to 72 % in the binary state of the streamwise velocity fluctuations in the logarithmic region is achieved; this corresponds to a correlation coefficient of $\approx$0.6. This thus demonstrates that wall-pressure sensing for velocity-state estimation – e.g. for use in real-time control of wall-bounded turbulence – has merit in terms of its realization at a range of Reynolds numbers.

Three different porous substrates (with different pore sizes, $s$, and permeabilities, $K$) are used to examine their effect on the structure of boundary layer flow over them. The flow is characterised with single-point hot-wire measurements as well as planar particle image velocimetry. In order to elucidate differences in shallow and deep flows past porous substrates, foams with two different thicknesses ($h$) are used (for all three substrates). A wide range of friction Reynolds numbers ($2000< Re_\tau <13\,500$) and permeability-based Reynolds numbers ($1< Re_K< 50$) are attained. For substrates with $Re_K \sim 1$, the flow behaviour remains similar to flow over impermeable smooth walls and as such Townsend's hypothesis remains valid. Very large-scale motions are observed over permeable foams even when the $Re_K > 1$. In contrast, a substantial reduction in velocity disturbances and associated length scales is achieved for permeable foams with intermediate values of pore density and relative foam thickness ($h/s$), which affects outer-layer similarity. As permeability is increased by increasing pore size, the foam becomes sparse relative to viscous scales at high Reynolds numbers. For such foams, the flow conforms to outer-layer similarity and is more akin to flow over rough surfaces. Permeability attenuates the wavelengths associated with the outer-layer peak.

In an effort to capture the continuous hydraulic jump and flow structure for a jet impinging on a disk, we recently proposed a composite mean-field thin-film approach consisting of subdividing the flow domain into three distinct connected regions of increasing gravity strength (Wang et al., J. Fluid Mech., vol. 966, 2023, A15). In the present study, we further validate our approach, and examine the characteristics and structure of the circular jump and recirculation. The influence of the disk radius is found to be significant, especially in the subcritical region. Below a disk radius, the jump transits from type Ia to type 0 after the recirculation zone has faded. The supercritical flow and jump location are insensitive to the disk size, but the jump length and height as well as the vortex size are strongly affected, all decreasing with decreasing disk radius, exhibiting a maximum with the flow rate for a small disk. The jump is relatively steep with a strong recirculation zone for a high obstacle at the disk edge. Comparison against the Navier–Stokes solution of Askarizadeh et al. (Phys. Rev. Fluids, vol. 4, 2019, 114002; Intl J. Heat Mass Transfer, vol. 146, 2020, 118823) for the weak and intermediate surface tension suggests that the surface tension effect is unimportant for a high obstacle for a jump of type 0 or type Ia. The film thickness at the disk edge for a freely draining film is found to comprise, in addition to a static component (capillary length), a dynamic component: ${h_\infty }\sim {(Fr/{r_\infty })^{2/3}}$ that we establish by minimizing the Gibbs free energy at the disk edge, and, equivalently, is also the consequence of the flow becoming supercritical near the edge. By assuming negligible film slope and curvature at the leading edge of the jump and maximum height at the trailing edge, we show that the jump length is related to the jump radius as ${L_J}\sim Re{(F{r^2}/{r_J}^5)^{1/3}}$. The vortex length follows the same behaviour. The energy loss and conjugate depth ratio exhibit a maximum with the flow rate, which we show to originate from the descending and ascending branches of the supercritical film thickness. The presence of the jump is not necessarily commensurate with that of a recirculation; the existence of the vortex closely depends on the upstream curvature and steepness of the jump. The surface separating the regions of existence/non-existence of the recirculation is given by the universal relation $R{e^{10/3}}F{r^2} = 9r_\infty ^9/50$. The jump can be washed off the edge of the disk, particularly at low viscosity and small disk size. The flow in the supercritical region remains insensitive to the change in gravity level and disk size but is greatly affected by viscosity.