With the growing interest in scale-resolving simulations of spatially evolving boundary layers, synthetic turbulence generation (STG) has become a valuable tool for providing unsteady turbulent boundary conditions through a sum over a finite number of spatio-temporal Fourier modes with amplitude, direction and phase determined by a random number set. Recent developments of STG methods are designed to match target profiles for anisotropic and inhomogeneous Reynolds stresses. In this paper, it is shown that, for practical values of the number of modes, a given set of random numbers may produce Reynolds stress profiles that are 30 % off their target. To remedy this situation, the error in the STG stress prediction is decomposed into a steady-state bias and a purely unsteady part affecting the time convergence. Direct relationships between the random number vectors and both types of error are developed, allowing large collections of random number sets to be rapidly scanned and the best performers selected for a much improved agreement with the target. The process is verified for the inflow to a direct numerical simulation of a flat plate at $Re_\theta = 1000$. This paper demonstrates sufficient time convergence over a few flow-through times as well as a correction of the method's biases.

The flow around a rotating circular cylinder has a parameter regime with a complex pattern of periodic solutions and multiple steady states. Sierra et al. (J. Fluid Mech., vol. 905, 2020, A2) provide a complete bifurcation analysis of this regime. The numerical computations are guided by a qualitative analysis of the bifurcations stemming from a highly degenerate singular dynamical system. Surprisingly, the dynamics of the singular system itself cannot be realized as a specific flow, but acts mathematically as an organizer of the physical bifurcation diagram.

This paper considers the effects of smoothly varying chordwise porosity of a finite perforated plate on turbulence–aerofoil interaction noise. The aeroacoustic model is made possible through the use of a novel Mathieu function collocation method, rather than a traditional Wiener–Hopf approach which would be unable to deal with chordwise-varying quantities. The main focus is on two bio-inspired porosity distributions, modelled from air flow resistance data obtained from the wings of barn owls (tyto alba) and common buzzards (buteo buteo). Trailing-edge noise is much reduced for the owl-like distribution, but, perhaps surprisingly, so too is leading-edge noise, despite both wings having similar porosity values at the leading edge. A general monotonic variation is then considered indicating that there may indeed be a significant acoustic impact of how the porosity is distributed along the whole chord of the plate, not just its values at the scattering edges. Through this investigation, it is found that a plate whose porosity continuously decreases from the trailing edge to a zero-porosity leading edge can, in fact, generate lower levels of trailing-edge noise than a plate whose porosity remains constant at the trailing-edge value.

In the flamelet regime of turbulent premixed combustion the enhancement in the burning rates originates primarily from surface wrinkling. In this work we investigate the Reynolds number dependence of burning rates of spherical turbulent premixed methane/air flames in decaying isotropic turbulence with direct numerical simulations. Several simulations are performed by varying the Reynolds number, while keeping the Karlovitz number the same, and the temporal evolution of the flame surface is compared across cases by combining the probability density function of the radial distance of the flame surface from the origin with the surface density function formalism. Because the mean area of the wrinkled flame surface normalized by the area of a sphere with radius equal to the mean flame radius is proportional to the product of the turbulent flame brush thickness and peak surface density within the brush, the temporal evolution of the brush and peak surface density are investigated separately. The brush thickness is shown to scale with the integral scale of the flow, evolving due to decaying velocity fluctuations and stretch. When normalized by the integral scale, the wrinkling scale defined as the inverse of the peak surface density is shown to scale with Reynolds number across simulations and as turbulence decays. As a result, the area ratio and the burning rate are found to increase as ${Re}_{\lambda }^{1.13}$, in agreement with recent experiments on spherical turbulent premixed flames. We observe that the area ratio does not vary with turbulent intensity when holding the Reynolds number constant.

A new class of self-propelled droplets, coined superwalkers, has been shown to emerge when a bath of silicone oil is vibrated simultaneously at a given frequency and its subharmonic tone with a relative phase difference between them (Valani et al., Phys. Rev. Lett., vol. 123, 2019, 024503). To understand the emergence of superwalking droplets, we explore their vertical and horizontal dynamics by extending previously established theoretical models for walkers driven by a single frequency to superwalkers driven by two frequencies. Here, we show that driving the bath at two frequencies with an appropriate phase difference raises every second peak and lowers the intermediate peaks in the vertical periodic motion of the fluid surface. This allows large droplets that could otherwise not walk to leap over the intermediate peaks, resulting in superwalking droplets whose vertical dynamics is qualitatively similar to normal walkers. We find that the droplet's vertical and horizontal dynamics are strongly influenced by the relative height difference between successive peaks of the bath motion, a parameter that is controlled by the phase difference. Comparison of our simulated superwalkers with the experiments of Valani et al. (2019) shows good agreement for small- to moderate-sized superwalkers.

The motion of a single spherical particle in a two-sided lid-driven cavity is investigated experimentally. The flow in which the particle moves is created by two facing cavity sidewalls which move with equal velocity in opposite directions. For a long cavity with width-to-height cross-sectional aspect ratio $\varGamma =W/H=1.6$ the flow field at Reynolds number ${Re}=400$ consists of steady spatially periodic three-dimensional convection cells. Nearly neutrally buoyant particles with radius in units of $H$ ranging from $1.1\times 10^{-2}$ to $7.1\times 10^{-2}$ are found to be attracted to periodic or quasi-periodic orbits in close vicinity of Kolmogorov–Arnold–Moser (KAM) tori of the unperturbed flow. Like the KAM tori the attractors of neutrally buoyant particles arise in mirror-symmetric pairs within each convection cell. The particle attractors are created by a dissipative effect in the dynamical system describing the particle motion which arises when the finite-size particle closely passes the moving walls. When the particle density deviates from that of the fluid, inertial attractors arise whose symmetry is broken by buoyancy, and other periodic attractors are created which do not have KAM tori as counterparts.

We present a study of droplet impingement on a two-layer liquid, specifically an oil droplet impinging on a layer of oil on water. In our experiments, the diameter and impact velocity of the droplet and the thickness of the oil layer were varied, and the maximum depth of the crater and the maximum height of the Worthington jet were measured. When the thickness of the oil layer was less than ${\sim }1.6$ times the droplet diameter, the depth of the crater depended on the thickness of the oil layer. Otherwise, the two-layer liquid behaved like a single layer. This observation is rationalized by considering the oil–water interface, whose deformation is negligible when the oil layer is thick but becomes significant when the oil layer is thinner. We define an effective Weber number for the two-layer liquid and show that the height of the Worthington jet is proportional to this effective Weber number.

The drying dynamics inside a network of interconnected channels driven by pervaporation, e.g. by diffusion of water through a permeable material surrounding the channels, is studied. The channels are initially filled with water and a single air/water meniscus is initiated at the entrance of the network; drying proceeds as menisci progressively invade the network. The study is focused on elementary networks: simple branched networks without reconnections, or simple loops, in order to get a clear physical picture on which an understanding of drying on more complex networks, such as those encountered in leaves, could be built in the near future. Experiments are compared with models which elaborate on a previously published single-channel model (Dollet et al., J. R. Soc. Interface, vol. 16, 2019, 20180690). In branched networks, experiments reveal velocity discontinuities of the menisci as they split at the nodes. In loops, it is found that the drying rate depends on the number of menisci bounding a given connected water region; when there are two such menisci, a prediction of the dynamics of each of them is proposed, based on the pervaporation-induced hydrodynamics inside the channels. Experiments and model predictions compare favourably for the global drying rate. Some deviations are found for the dynamics of individual menisci, which are ascribed to the sensitivity of the dynamics to small fluctuations in wetting conditions.

Both experiments and direct numerical simulations have been used to demonstrate that riblets can reduce turbulent drag by as much as $10\,\%$, but their systematic design remains an open challenge. In this paper we develop a model-based framework to quantify the effect of streamwise-aligned spanwise-periodic riblets on kinetic energy and skin-friction drag in turbulent channel flow. We model the effect of riblets as a volume penalization in the Navier–Stokes equations and use the statistical response of the eddy-viscosity-enhanced linearized equations to quantify the effect of background turbulence on the mean velocity and skin-friction drag. For triangular riblets, our simulation-free approach reliably predicts drag-reducing trends as well as mechanisms that lead to performance deterioration for large riblets. We investigate the effect of height and spacing on drag reduction and demonstrate a correlation between energy suppression and drag reduction for appropriately sized riblets. We also analyse the effect of riblets on drag-reduction mechanisms and turbulent flow structures including very large-scale motions. Our results demonstrate the utility of our approach in capturing the effect of riblets on turbulent flows using models that are tractable for analysis and optimization.

Inertia-dominated wall-sheared turbulent flows are composed of an inner and outer layer, where the former is occupied by the well-known autonomous inner cycle while the latter is composed of coherent structures with spatial extent comparable to the flow depth. In arbitrary streamwise–wall-normal planes, outer-layer structures instantaneously manifest as regions of quasi-uniform momentum – relative excesses and deficits about the Reynolds average – and for this reason are termed uniform momentum zones (UMZs). By virtue of this attribute, the interfacial zones between successive UMZs exhibit abrupt wall-normal gradients in streamwise momentum; these interfacial gradients cannot be explained by the notion of attached eddies, for which the vertical gradient goes as $(x_3^+)^{-1}$ in the outer layer, where $x_3^+$ is inner-normalized wall-normal position. Using data from direct numerical simulation (DNS) of channel turbulence across inertial regimes, we recover vertical profiles of Kolmogorov length a posteriori and show that $\eta ^+ \sim (x_3^+)^{1/4}$, thereby requiring that ambient wall-normal gradients in streamwise velocity must scale as $(x_3^+)^{-1/2}$. The data reveal that UMZ interfaces are responsible for these relatively larger wall-normal gradients. The DNS data afford a unique opportunity to interpret inner- and outer-layer structures simultaneously: we propose that UMZs – and the associated outer-layer dynamics – can be explained as the product of inner-layer bluff-body-like interactions, wherein wakes of quasi-uniform momentum emanate from the inner layer; wake-scaling arguments agree with observations from DNS.

We consider the trailing-edge vortex produced in an inviscid fluid by the start-up motion of a two-dimensional flat plate. A general starting motion is studied that includes the initial angle-of-attack of the plate (which may be zero), individual time power laws for plate translational and rotational speeds and the pivot position for plate rotation. A vortex-sheet representation for a start-up separated flow at the trailing edge is developed whose time-wise evolution is described by a Birkhoff–Rott equation coupled to an appropriate Kutta condition. This description includes convection by the outer flow, rotation and vortex-image self-induction. It admits a power-law similarity solution for the (small-time) primitive vortex, leading to an equation set where each term carries its own time-wise power-law factor. A set of four general plate motions is defined. Dominant-balance analysis of this set leads to discovery of three distinct start-up vortex-structure types that form the basis for all vortex motion. The properties of each type are developed in detail for some special cases. Numerical and analytical solutions are described and transition between solution types is discussed. Singular and degenerate vortex behaviour is discovered which may be due to the absence of fluid viscosity. An interesting case is start-up motion with zero initial angle of attack coupled to power-law plate rotation for which time-series examples are given that can be compared to high Reynolds number viscous flows.

Preferential flow that leads to non-uniform displacement, especially in heterogeneous porous media, is usually unwelcome in most practical processes. We propose a self-adaptive preferential flow control mechanism by using dispersed polymers, which is supported strongly by experimental and numerical evidence. Our experiments are performed on a microchip with heterogeneous porous structures where oil is displaced by dispersed polymer microsphere particles. Even though the size of the particles is much smaller than the pore-throat size, the diversion effect by the dispersed microspheres is still proved. Therefore, the plugging effect is not the major mechanism for preferential flow control by dispersed polymers. The mechanisms are further investigated by pore-scale modelling, which indicates that the dispersed polymers exhibit an adaption ability to pressure and resistance in the porous flow field. In such an intelligent way, the displacing fluid with dispersed polymers smartly controls the preferential flow by inducing pressure fluctuations, and demonstrates better performance in both efficiency and economic aspects than the traditional method by simply increasing the viscosity. These insights can be applied to improve techniques in the field, such as enhanced oil recovery and soil wetting.

In nature, the wind waves of the gravity–capillary range are noticeably skewed forward. The salient feature of such waves is a characteristic pattern of capillary ripples on their crests. The train of these ‘parasitic capillaries’ is not symmetric with respect to the crest, it is localised on the front slope and decays towards the trough. Although understanding the gravity–capillary waves front–back asymmetry is important for remote sensing and, potentially, for wave–wind interaction, the physical mechanisms causing this asymmetry have not been identified. Here, we address this gap by extensive numerical simulations of the Euler equations employing the method of conformal mapping for two-dimensional potential flow and taking into account wave generation by wind and dissipation due to molecular viscosity. On examining the role of various factors contributing to the wave profile front–back asymmetry: wind forcing, viscous stresses and the Reynolds stresses caused by ripples, we found, in the absence of wave breaking, the latter to be by far the most important. It is the lopsided ripple distribution which leads to the noticeable fore–aft asymmetry of the mean wave profile. We also found how the asymmetry depends on wavelength, steepness, wind, viscosity and surface tension. The results of the model are discussed in the context of the available experimental data on asymmetry of gravity–capillary waves in both the breaking and non-breaking regimes. A reasonable agreement of the model with the data has been found for the regime without breaking or microbreaking.

Stability analysis of a liquid layer subjected to an oblique temperature gradient (OTG) is carried out. The general linear stability analysis reveals a stabilization effect of the imposed horizontal component (horizontal temperature gradient, HTG) of the OTG on the long-wave instabilities introduced by the vertical component (vertical temperature gradient, VTG) of the OTG. This stabilization is due to the VTG induced by the prescribed HTG, which counteracts the imposed VTG. The induced VTG arises due to the presence of advection of the energy. As a result of the stabilization, the long-wave mode forms an island of instability in the $\eta$–$Ma_c$ plane, where $\eta$ and $Ma_c$ are the ratio of the strength of the imposed HTG to imposed VTG components of the OTG, and the critical Marangoni number, respectively. However, for sufficiently high $\eta$, a new class of modes emerge with the critical Marangoni number scaling as $Ma_c \sim 1/\eta$. These modes originate as a result of the interaction between the thermocapillary flow caused by the imposed HTG on the one hand, and the VTG on the other, and remain the dominant modes of instability at higher $\eta$. The long-wave analysis is carried out and, in its framework, the nonlinear evolution equation is derived, and, based on it, linear and weakly nonlinear analyses are performed. An increase in $\eta$ changes the type of bifurcation from subcritical to supercritical. The numerical solution of the evolution equation around the critical parameter values validates the predictions of the weakly nonlinear analysis. The present study illustrates a possible use of imposing the HTG to prevent dry-spot formation and rupture of the film caused by the imposed VTG.

The self-excited instabilities acting on laminar separation bubbles in the absence of external forcing are studied by means of linear stability analysis and direct numerical simulation. Previous studies demonstrated the existence of a three-dimensional modal instability, that becomes active for bubbles with peak reversed flow of approximately $7\,\%$ of the free-stream velocity, well below the ${\approx } 16\,\%$ required for the absolute instability of Kelvin–Helmholtz waves. Direct numerical simulations are used to describe the nonlinear evolution of the primary instability, which is found to correspond to a supercritical pitchfork bifurcation and results in fully three-dimensional flows with spanwise inhomogeneity of finite amplitude. An extension of the classic weakly non-parallel analysis is then applied to the bifurcated flows, that have a strong dependence on the cross-stream planes and a mild dependence on the streamwise direction. The spanwise distortion of the separated flow induced by the primary instability is found to strongly destabilize the Kelvin–Helmholtz waves, leading to their absolute instability and the appearance of a global oscillator-type instability. This sequence of instabilities triggers the laminar–turbulent transition without requiring external disturbances or actuation. The characteristic frequency and streamwise and spanwise wavelengths of the self-excited instability are in good agreement with those reported for low-turbulence wind-tunnel experiments without explicit forcing. This indicates that the inherent dynamics described by the self-excited instability can also be relevant when external disturbances are present.

The role of large-scale motions (LSMs) in energy transfer is investigated by analysing wall-parallel velocity fields at low-to-moderate Reynolds number ($Re_{\tau }=1200\text {--}3500$), which are obtained via a two-dimensional (2-D) particle image velocimetry measurement with large field-of-view. Two types of energy flux, i.e. local interscale energy flux and in-plane spatial energy flux are inspected in detail. Targeting the energy transfer in large-scale regime, an anisotropic filter is designed based on the zero-crossing scale boundary in a 2-D energy transfer spectrum, across which the net energy flux is the maximum. This ‘optimal’ energy flux boundary separates the scale space into an energy donating large-scale part and an energy receiving small-scale one. The crossover energy flux, as well as the associated flow field structures, are studied by conditional statistics and linear stochastic estimation, in which the statistical spanwise symmetry is deliberately broken by designing special velocity gradient conditions for event probing. A strong connection between large-scale energy flux events and LSMs are found. Namely, forward scatter events have higher probability to reside on the wavy flank of low-momentum LSMs, if compared with the scenario of being clamped in the middle of two streamwise-aligned high- and low-momentum LSMs (Natrajan & Christensen, Phys. Fluids, vol. 18, issue 6, 2006, pp. 299–325). Meanwhile, pairs of positive and negative spatial transfer events tend to locate inside LSMs. It is thus argued that the meandering nature of LSMs, which forms the necessary velocity gradient, might play a determining role in the process of large-scale energy transfer. The spatial correlation between them is then schematized in a conceptual model, which explains most of the present observations.

Analytical solutions are presented for longitudinal flow along a superhydrophobic annular pipe where one wall, either the inner or outer, is a fully no-slip boundary while the other is a no-slip wall decorated by a rotationally symmetric pattern of no-shear longitudinal stripes. Formulas are given for the effective slip length associated with laminar flow along the superhydrophobic pipe and the friction properties are characterized. It is shown how these new solutions generalize two solutions to mixed no-slip/no-shear boundary value problems due to Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372) for flow in a single-walled superhydrophobic pipe and a superhydrophobic channel. This is done by providing alternative representations of Philip's two solutions, including a useful new formula for the effective slip length for his channel flow solution. For a superhydrophobic annular pipe with inner-wall no-shear patterning there is an optimal pipe bore for enhancing hydrodynamic slip for a given pattern of no-shear stripes. These optimal pipes have a ratio of inner–outer pipe radii in the approximate range 0.5–0.6 and depending only weakly on the geometry of the surface patterning. Boundary point singularities are found to be deleterious to the slip suggesting that, in designing slippery pipes, maximizing the size of uninterrupted no-shear regions is preferable to covering the same surface area with a larger number of smaller no-shear zones. The results add to a list of analytical solutions to mixed boundary value problems relevant to modelling superhydrophobic surfaces.

We study the effect of insoluble surfactants on the wave dynamics of vertically falling liquid films. We use three-dimensional numerical simulations and employ a hybrid interface-tracking/level-set method, taking into account Marangoni stresses induced by gradients of interfacial surfactant concentration. Our numerical predictions for the evolution of the surfactant-free, three-dimensional wave topology are validated against the experimental work of Park & Nosoko (AIChE J., vol. 49, 2003, pp. 2715–2727). The addition of surfactants is found to influence significantly the development of horseshoe-shaped waves. At low Marangoni numbers, we show that the wave fronts exhibit spanwise oscillations before eventually acquiring a quasi-two-dimensional shape. In addition, the presence of Marangoni stresses is found to suppress the peaks of the travelling waves and preceding capillary wave structures. At high Marangoni numbers, a near-complete rigidification of the interface is observed.

We investigate the reconstruction of a turbulent flow field in the atmospheric boundary layer from a time series of lidar measurements, using large-eddy simulations (LES) and a four-dimensional variational data assimilation algorithm. This leads to an optimisation problem in which the error between measurements and simulations is minimised over an observation time horizon. We also consider reconstruction based on a Taylor's frozen turbulence (TFT) model as a point of comparison. To evaluate the approach, we construct a series of virtual lidar measurements from a fine-grid LES of a pressure-driven boundary layer. The reconstruction uses LES on a coarser mesh and smaller domain, and results are compared to the fine-grid reference. Two lidar scanning modes are considered: a classical plan-position-indicator mode, which swipes the lidar beam in a horizontal plane, and a three-dimensional pattern that is based on a Lissajous curve. We find that normalised errors lie between $15\,\%$ and $25\,\%$ (error variance normalised by background variance) in the scanning region, and increase to $100\,\%$ over a distance that is comparable to the correlation length scale outside this scanning region. Moreover, LES outperforms TFT by 30 %–70 % depending on scanning mode and location.

The destabilization of a dielectric film flow due to an electrostatic surface force is investigated. A weighted residuals integral boundary-layer (WIBL) model is derived and validated against full numerical simulations. The equations of the WIBL model indicate that the electrostatic surface force contributes to the evolution equations in a similar mathematical way as the volumetric gravitational force. Contrary to gravity, an additional electrostatic contribution ($\chi _2$) arises, whose impact increases nonlinearly with decreasing capacitor plate distance. This nonlinear contribution causes a fold of the branch of solutions of the dynamical system and, thus, the co-existence of a low amplitude solution that is stable against infinitesimal disturbances and an unstable high amplitude solution. In time-dependent simulations, the fold coincides with the limit in the parameter space beyond which a finite-time blow-up occurs with an unsaturated growth of the main wave hump leading to wave pinch-off and drop formation. Thus, a phase diagram can be constructed by tracking this fold. The shape of the main wave prior to blow-up depends on the electrostatic parameter $\chi _2$. If this parameter is zero, the force is equivalent to a hanging film flow configuration and dripping occurs with a drop-shaped structure. With an increasing contribution of the parameter $\chi _2$, Taylor-cone waves occur prior to finite-time blow-up, leading to jetting. Finally, the transition from stable to unstable waves is investigated in terms of the two dimensionless electric parameters, the Reynolds, and viscous dissipation numbers. Imposing the most amplified wavelength, a transition border between stable solutions and jetting is identified.