This paper is concerned with the boundary layer on the leading edge of an aerofoil with the aerofoil surface sliding parallel to itself in the upstream direction. The flow analysis is conducted in the framework of the classical Prandtl formulation with the pressure distribution given by the solution for the outer inviscid flow. Since a reverse flow region is always present near the wall, a numerical method, where the derivatives were approximated by the windward finite differences, was used to solve the boundary-layer equations. We were interested in the flow behaviour on the upper surface of the aerofoil, but to calculate the boundary-layer equations, we had to extend the computational domain from the upper surface of the aerofoil to the lower surface. The calculations were performed for a range of angles of attack, and it is found that there exists a critical value of the angle of attack for which the Moore–Rott–Sears singularity forms in the flow. This is accompanied by an abrupt thickening of the boundary layer at the singular point and the formation of a recirculation region with closed streamlines behind this point. We further found that the flow immediately behind the singular point and in the recirculation region could be treated as inviscid, which allowed us to use the Prandtl–Batchelor theorem for theoretical modelling of the flow. A similar formulation was used earlier by Bezrodnykh et al. (Comput. Maths Math. Phys. vol. 63, 2023, pp. 2359–2371). These authors considered the boundary-layer flow on a flat plate with the pressure gradient created by a dipole situated some distance from the plate. They also found that there exists a critical value of the dipole strength for which a singularity forms in the boundary layer. However, their interpretation of the flow behaviour differs significantly from what we observe in our study.

Immiscible two-phase flows in geological fractures are relevant to various industrial applications, including subsurface fluid storage and hydrocarbon exploitation. Direct numerical simulations (DNS) of first-principle equations, which resolve three-dimensional (3-D) fluid–fluid interfaces, can address all types of flow regimes but are computationally intensive. To retain most of their advantages while reducing the computational cost, we propose a novel two-dimensional (2-D) model based on integrating the 3-D first-principle equations over the local fracture aperture, assuming the lubrication approximation and a parabolic out-of-plane velocity profile, and relying on the volume-of-fluid method for fluid–fluid interface capturing. Such existing models have, so far, been restricted to single-phase permanent flow in rough fractures and two-phase flow in 2-D porous media. Wall friction and out-of-plane capillary pressure are incorporated as additional terms in the 2-D momentum equation. The model then relies on a geometric description reduced to the fracture’s aperture field and mean topography field. Implemented in OpenFOAM, it is validated against 3-D DNS results for viscous fingering in a Hele-Shaw cell, and applied to a realistic synthetic rough fracture geometry over a wide range of capillary numbers ($Ca$). We then analyse to which extent, under which conditions and why this depth-integrated 2-D model, with a tenfold reduction in computational cost, provides convincing results compared with 3-D DNS predictions. We find that it performs surprisingly well over nearly the entire range of $Ca$ for which 3-D DNS models are relevant, in particular because it properly accounts for the out-of-plane capillary forces and wall friction.

The impact of several ‘flavours’ of free-stream turbulence (FST) on the structural response of a cantilever cylinder, subjected to a turbulent cross-flow is investigated. At high enough Reynolds numbers, the cylinder generates a spectrally rich turbulent wake that contributes significantly to the experienced loads. The presence of FST introduces additional complexity through two primary mechanisms: directly, by imposing a fluctuating velocity field on the cylinder’s surface, and indirectly, by altering the vortex shedding dynamics, modifying the experienced loads. We employ concurrent temporally resolved particle image velocimetry and distributed strain measurements using Rayleigh backscattering fibre optic sensors to instrument the surrounding velocity field and the structural strain respectively. By using various turbulence-generating grids, and manipulating their distance to the cylinder, we assess a broad FST parameter space allowing us to explore individually the influence of the transverse integral length scale ($\mathcal{L}_{13}/D$) and turbulence intensity of the FST on the developing load dynamics. The FST enhances the magnitude of the loads acting on the cylinder. This results from a decreased vortex formation length, increased coherence of regular vortex shedding, and energy associated with this flow structure in the near wake. The cylinder’s structural response is driven mainly by the vortex shedding dynamics, and its modification induced by the presence of FST, i.e. the indirect effect outweighs the direct effect. From the explored FST parameter space, turbulence intensity was seen to be the main driver of enhanced loading conditions, presenting a positive correlation with the fluctuating loads magnitude at the root.

Viscous fingering, a classic hydrodynamic instability, is governed by the the competition between destabilising viscosity ratios and stabilising surface tension or thermal diffusion. We show that the channel confinement can induce ‘diffusion’-like stabilising effects on viscous fingering even in the absence of interfacial tension and thermal diffusion, when a clear oil invades the mixture of the same oil and non-colloidal particles. The key lies in the generation of long-range dipolar disturbance flows by highly confined particles that form a monolayer inside a Hele-Shaw cell. We develop a coarse-grained model whose results correctly predict universal fingering dynamics that is independent of particle concentrations. This new mechanism offers insights into manipulating and harnessing collective motion in non-equilibrium systems.

We present an experimental study on the effects of polymer additives on the turbulent/non-turbulent interface (TNTI) in a fully developed round water jet. The Reynolds number based on the jet diameter is fixed at $Re=7075$. The Weissenberg number $Wi$ ranges from 24 to 86. We employ time-resolved simultaneous particle image velocimetry and laser-induced fluorescence measurements to investigate the local entrainment and engulfment process along the TNTI in two regimes: entrainment transition and enhancement regimes. In polymer-laden jets, the TNTI fluctuates more intermittently in the radial direction and more ambient fluid can be engulfed into the turbulent region due to the augmented large scale motion. Though the contribution of engulfment to the total flux increases with $Wi$, engulfment is still not the major contribution to the entrainment in polymer-laden jets. We further show that the local entrainment velocity is increased in both regimes compared with the pure water jet, due to two contributions: polymer elastic stress and the more intermittent character of the TNTI. In the entrainment transition regime, we observe smaller fractal dimension and shorter length of TNTI compared with the Newtonian case, consistent with previous numerical simulations (Abreu et al. J. Fluid Mech. vol. 934, 2022, A36); whereas those in the enhancement regime remain largely unchanged. The difference between the two regimes results from the fact that the jet flow decays in the streamwise direction. In the entrainment transition regime, turbulence intensity is strong enough to significantly stretch the polymers, resulting in a smoother TNTI in the inertial range. However, in the entrainment enhancement regime, the polymer’s feedback is not strong enough to alter the fractal dimension due to the low elasticity. The above mentioned differences of entrainment velocity and TNTI in the entrainment reduction/transition and enhancement regimes also explain the reduced and enhanced spreading rate of the viscoelastic jet observed in previous numerical simulations and experiments (Guimarães et al. J. Fluid Mech. 2020,vol. 899, A11; Peng et al. Phys. Fluids, 2023, vol. 35, 045110).

From anthropogenic litter carried by ocean currents to plant stems travelling through the atmosphere, geophysical flows are often seeded with elongated, fibre-like particles. In this study, we used a large-scale laboratory model of a tidal current – representative of a widespread class of geophysical flows – to investigate the tumbling motion of long, slender and floating fibres in the complex turbulence generated by flow interactions with a tidal inlet. Despite the non-stationary, non-homogeneous and anisotropic nature of this turbulence, we find that long fibres statistically rotate at the same frequency as eddies of similar size, a phenomenon called scale selection, which is known to occur in ideal turbulence. Furthermore, we report that the signal of the instantaneous transverse velocity difference between the fibre ends changes significantly from the signal produced by the flow in the fibre surroundings, although the two are statistically equivalent. These observations have twofold implications. On the one hand, they confirm the reliability of using the end-to-end velocity signal of rigid fibres to probe the two-point transverse statistics of the flow, even under realistic conditions: oceanographers could exploit this observation to measure transverse velocity differences through elongated floats in the field, where superdiffusion complicates collecting sufficient data to probe two-point turbulence statistics at a fixed separation effectively. On the other hand, by addressing the dynamics of inertial range particles floating in the coastal zone, these observations are crucial to improving our ability to predict the fate of meso- and macro-litter, a size class that is currently understudied.

We study theoretically and experimentally the propagation of two bubbles in a Hele-Shaw cell under a uniform background flow. We consider the regime where the bubbles are large enough to be flattened by the cell walls into a pancake-like shape, but small enough such that each bubble remains approximately circular when viewed from above. In a system of two bubbles of different radii, if the smaller bubble is in front, it will be overtaken by the larger bubble. Under certain circumstances, the bubbles may avoid collision by rolling over one another while passing. We find that, for a given ratio of the bubble radii, there exists a critical value of a dimensionless parameter (the Bretherton parameter) above which the two bubbles will never collide, regardless of their relative size and initial transverse offset, provided they are initially well separated in the direction of the background flow. Additionally, we determine the corrections to the bubble shape from circular for two bubbles aligned with the flow direction. We find that the front bubble flattens in the flow direction, while the rear bubble elongates. These shape changes are associated with changes in velocity, which allow the rear bubble to catch the bubble in front even when they are of the same size.

A previously developed modelling procedure for large eddy simulations (LESs) is extended to allow physical space implementations for inhomogeneous flows. The method is inspired by the well-established theoretical analyses and numerical investigations of homogeneous isotropic turbulence. A general procedure that focuses on recovering the full subgrid scale (SGS) dissipation from resolved fields is formulated, combining the advantages of both the structural and the functional strategy of SGS modelling. The interscale energy transfer is obtained from the test-filtered velocity field, corresponding to the subfilter scale (SFS) stress, or, equivalently, the similarity model is used to compute the total SGS dissipation. The energy transfer is then cast in the form of eddy viscosity, allowing the method to retain the desired total SGS dissipation in low resolution LES runs. The procedure also exhibits backscatter without causing numerical instabilities. The new approach is general and self-contained, working well for different filtering kernels, Reynolds numbers and grid resolutions.