Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers ${ {Ek}} \gtrsim 10^{-8}$), while realistic ${Ek}$ for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($ {Ek}\to 0$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite ${Ek}$ has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for ${Ek}\to 0$, recently introduced by Julien et al. (2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to $ {Ek}=10^{-15}$ and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers (${Ek}\approx 10^{-9}$). We also identify an overshoot in the heat transport as ${Ek}$ is varied at fixed $\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$, where $Ra$ is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small ${Ek}$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.

Previous studies on the scaling of pressure fluctuations in wall-bounded turbulent flows have typically employed the same frameworks as those used for mean flow, with inner scaling based on frictional velocity and viscous length scales, and outer scaling relying on boundary layer thickness or displacement thickness. These traditional scales primarily reflect the characteristics of the mean streamwise velocity profile and momentum balance. In this work, we propose novel scaling frameworks for pressure fluctuations in turbulent channel and pipe flows, derived from the Poisson equation for pressure fluctuations. Applying the scaling patch approach, we analyse the rapid and slow terms in the Poisson equation, and introduce new scaling for pressure fluctuation variance in both the inner and outer regions. These new scales are designed to better capture the influence of Reynolds stresses by incorporating their peak values. Additionally, we establish a strong correlation between the root mean square (r.m.s.) of pressure fluctuations and the Reynolds shear stress, resulting in an empirical equation that accurately predicts their ratio. This equation provides a practical method for estimating the r.m.s. of pressure fluctuations in the flow, which remains challenging to measure in experimental investigations.

In this paper, we discuss the transport of sediment and the formation of bedforms in turbulent river flows, under flow conditions typical of flooding events. Through the implementation of an immersed boundary method, a wall model and a morphological model, we were able to simulate complex and mobile geometries under high Reynolds numbers at an affordable computational cost. In particular, we examined the evolution of bedforms on a loose sediment bed under turbulent flow conditions, using input parameters obtained from laboratory measurements. Over time, the bedforms become more three-dimensional and irregular in shape, leading to changes in the shear layer, crest angle and separation patterns. The bedforms continue to evolve until a quasi-steady equilibrium is reached. Our simulations highlight the crucial role played by the small-scale bedforms, which significantly affect the flow dynamics: an increase in the total drag is observed, related to the form drag generated by the local recirculation and the increased size of the large-scale recirculation bubble. Furthermore, a stronger turbulent activity ensues from the shear layers forming on the crests of the small-scale bedforms. Finally, a wider shedding angle of the shear layer is caused by the irregular crest line.

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared with the channel height and the Péclet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk–surface exchange. In scenarios with strong exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.

Carbon storage in saline aquifers is a prominent geological method for reducing CO2 emissions. However, salt precipitation within these aquifers can significantly impede CO2 injection efficiency. This study examines the mechanisms of salt precipitation during CO2 injection into fractured matrices using pore-scale numerical simulations informed by microfluidic experiments. The analysis of varying initial salt concentrations and injection rates revealed three distinct precipitation patterns, namely displacement, breakthrough and sealing, which were systematically mapped onto regime diagrams. These patterns arise from the interplay between dewetting and precipitation rates. An increase in reservoir porosity caused a shift in the precipitation pattern from sealing to displacement. By incorporating pore structure geometry parameters, the regime diagrams were adapted to account for varying reservoir porosities. In hydrophobic reservoirs, the precipitation pattern tended to favour displacement, as salt accumulation occurred more in larger pores than in pore throats, thereby reducing the risk of clogging. The numerical results demonstrated that increasing the gas injection rate or reducing the initial salt concentration significantly enhanced CO2 injection performance. Furthermore, identifying reservoirs with high hydrophobicity or large porosity is essential for optimising CO2 injection processes.

Periodic gravity-capillary waves on a fluid of finite depth with constant vorticity are studied theoretically and numerically. The classical Stokes expansion method is applied to obtain the wave profile and the interior flow up to the fourth order of approximation, which thereby extends the works of Barakat & Houston (1968) J. Geophys. Res. 73 (20), 6545–6554 and Hsu et al. (2016) Proc. R. Soc. Lond. A 472, 20160363. The classical perturbation scheme possesses singularities for certain wavenumbers, whose variations with depth are shown to be affected by the vorticity. This analysis also reveals that for any given value of the physical depth, there exists a threshold value of the vorticity above which there are no singularities in the theoretical solution. The validity of the third- and fourth-order solutions is examined by comparison with exact numerical results, which are obtained with a method based on conformal mapping and Fourier series expansions of the wave surface. The outcomes of this comparison are surprising as they report important differences in the internal flow structure, when compared with the third-order predictions, even though both approximations predict almost perfectly the phase velocity and the surface profiles. Usually, this occurs when the wavenumber is far enough from a critical value and the steepness is not too large. In these non-resonant cases, it is found that the fourth-order theory is more consistent with the exact numerical results. With negative vorticity the improvement is noticeable both beneath the crest and the trough, whereas with positive vorticity the fourth-order theory does a better job either beneath the crest or beneath the trough, depending of the type of the wave.

Two-dimensional simulations incorporating detailed chemistry are conducted for detonation initiation induced by dual hot spots in a hydrogen/oxygen/argon mixture. The objective is to examine the transient behaviour of detonation initiation as facilitated by dual hot spots, and to elucidate the underlying mechanisms. Effects of hot spot pressure and distance on the detonation initiation process are assessed; and five typical initiation modes are identified. It is found that increasing the hot spot pressure promotes detonation initiation, but the impact of the distance between dual hot spots on detonation initiation is non-monotonic. During the initiation process, the initial hot spot autoignites, and forms the cylindrical shock waves. Then, the triple-shock structure, which is caused by wave collisions and consists of the longitudinal detonation wave, transverse detonation wave and cylindrical shock wave, dominates the detonation initiation behaviour. A simplified theoretical model is proposed to predict the triple-point path, whose curvature quantitatively indicates the diffraction intensity of transient detonation waves. The longitudinal detonation wave significantly diffracts when the curvature of the triple-point path is large, resulting in the failed detonation initiation. Conversely, when the curvature is small, slight diffraction effects fail to prevent the transient detonation wave from developing. The propagation of the transverse detonation wave is affected not only by the diffraction effects but also by the mixture reactivity. When the curvature of the triple-point trajectory is large, a strong cylindrical shock wave is required to compress the mixture, enhancing its reactivity to ensure the transverse detonation wave can propagate without decoupling.

The Edgerton crown is an iconic manifestation of drop impact splashing, with its prominent cylindrical edge decorated with detaching droplets. Herein, we identify the formation of an intriguing double-crown, when a high-viscosity drop impacts on a shallow pool of a lower-viscosity immiscible liquid. High-speed imaging shows that after the initial fine horizontal ejecta sheet, the first inner crown emerges vertically from the film liquid. This is followed by the second crown which forms near the outer base of the first crown, as the tip of the horizontally spreading viscous drop approaches the outer free surface. Axisymmetric numerical simulations, using the volume-of-fluid method with adaptive grid refinement, show that the flow squeezed out between the viscous drop and the solid surface, generates two counter-rotating vortex rings, which travel radially outwards together and drive out the second crown through the free surface. The bottom vortex emerges from the separated boundary layer at the solid wall, while the top one detaches from the underside of the viscous drop. We map out the narrow parameter regime, where this ephemeral structure emerges, in terms of viscosity ratio, impact velocity and film thickness.

We investigate the drag reduction effects by two representative blowing/suction-based control methods having different drag reduction mechanisms, i.e. the opposition control and uniform blowing (UB), in a bump-installed turbulent channel flow through direct numerical simulations. We consider two different bulk Reynolds numbers ${\textit {Re}}_b = 5600$ and $12\,600$, and bump heights $h^+ \approx 20$ and $40$. In the opposition-controlled case, the friction drag reduction effect in the case with a bump is similar to that in the case without a bump, while the control effect on the pressure drag is hardly observed. The total drag reduction rate decreases for the higher bump height because the ratio of the pressure drag to the total drag increases as the bump height. In the UB case, UB at $0.1\,\%$ or $0.5\,\%$ of the bulk-mean velocity is imposed on the lower wall with a bump, while the same amount of uniform suction (US) is applied on the upper flat wall to keep the mass flow rate. Although the total friction drag increases due to a detrimental effect of US on the upper wall, the wall-normal motions due to the existence of a bump on the lower wall are suppressed by the UB, so that the pressure drag is decreased, unlike the opposition-controlled case. Due to the difference in the inherent drag reduction mechanisms, the flow separation in the region behind the bump is enhanced by the opposition control, while suppressed by UB.

The critical points of vorticity in a two-dimensional viscous flow are essential for identifying coherent structures in the vorticity field. Their bifurcations as time progresses can be associated with the creation, destruction or merging of vortices, and we analyse these processes using the equation of motion for these points. The equation decomposes the velocity of a critical point into advection with the fluid and a drift proportional to viscosity. Conditions for the drift to be small or vanish are derived, and the analysis is extended to cover bifurcations. We analyse the dynamics of vorticity extrema in numerical simulations of merging of two identical vortices at Reynolds numbers ranging from 5 to 1500 in the light of the theory. We show that different phases of the merging process can be identified on the basis of the balance between advection and drift of the critical points, and identify two types of merging, one for low and one for high values of the Reynolds number. In addition to local maxima of positive vorticity and minima of negative vorticity, which can be considered centres of vortices, minima of positive vorticity and maxima of negative vorticity can also exist. We find that such anti-vortices occur in the merging process at high Reynolds numbers, and discuss their dynamics.

The hydrodynamic interactions between a sedimenting microswimmer and a solid wall have ubiquitous biological and technological applications. A plethora of gravity-induced swimming dynamics near a planar no-slip wall provide a platform for designing artificial microswimmers that can generate directed propulsion through their translation–rotation coupling near a wall. In this work, we provide exact solutions for a squirmer (a model swimmer of spherical shape with a prescribed slip velocity) facing either towards or away from a planar wall perpendicular to gravity. These exact solutions are used to validate a numerical code based on the boundary integral method with an adaptive mesh for distances from the wall down to 0.1 % of the squirmer radius. This boundary integral code is then used to investigate the rich gravity-induced dynamics near a wall, mapping out the detailed bifurcation structures of the swimming dynamics in terms of orientation and distance to the wall. Simulation results show that a squirmer may traverse the wall, move to a fixed point at a given height with a fixed orientation in a monotonic way or in an oscillatory fashion, or oscillate in a limit cycle in the presence of wall repulsion.

In the present study, we investigate the modulation effects of particles on compressible turbulent boundary layers at a Mach number of 6, employing high-fidelity direct numerical simulations based on the Eulerian–Lagrangian point-particle approach. Our findings reveal that the mean and fluctuating velocities in particle-laden flows exhibit similarities to incompressible flows under compressibility transformations and semi-local viscous scaling. With increasing particle mass loading, the reduction in Reynolds shear stress and the increase in particle feedback force constitute competing effects, leading to a non-monotonic variation in skin friction, particularly in turbulence over cold walls. Furthermore, dilatational motions near the wall, manifested as travelling-wave structures, persist under the influence of particles. However, these structures are significantly weakened due to the suppression of solenoidal bursting events and the negative work exerted by the particle feedback force. These findings align with the insight of Yu et al. (J. Fluid. Mech., vol. 984, 2024, A44), who demonstrated that dilatational motions are generated by the vortices associated with intense bursting events, rather than acting as evolving perturbations beneath velocity streaks. The attenuation of travelling-wave structures at higher particle mass loadings also contributes to the reduction in the intensities of wall shear stress and heat flux fluctuations, as well as the probability of extreme events. These results highlight the potential of particle-laden flows to mitigate aerodynamic forces and thermal loads in high-speed vehicles.

We analyse the small-scale characteristics, such as enstrophy, total strain and normality/non-normality, in the three-dimensional, separated flow around a NACA 0018 wing using direct numerical simulations. The angle of attack is $10^\circ$ and the Reynolds number (based on the chord length) is $Re_c=5000$. The role of non-normality is investigated by performing Schur decomposition of the velocity gradient tensor. We also apply the Schur decomposition to derive new expressions for the production of enstrophy and total strain arising from the mean flow inhomogeneity. We focus on two sections of the flow, across the recirculating zone and along the transitioning shear layer, and compare our results with homogeneous isotropic turbulence (HIT). Within the recirculating region, the non-normality index is approximately 0 (and close to the HIT value), indicating almost equal normal and non-normal contributions. However, in the separating layer non-normal effects strongly dominate, especially in the region of kinetic energy growth. Only in the decay region do the values of the non-normality index gradually approximate HIT values. The production of enstrophy due to vortex stretching is dominated by the mixed (interaction) term, where normal strain stretches non-normal vorticity. The same component also dominates the strain self-amplification term. The contributions of different QR regions to the production terms are also examined. Production due to mean strain rate is triggered upstream compared with production due to fluctuating strain fields.

The settling velocity of frozen hydrometeors in the atmospheric surface layer depends on their inertial and drag properties, and on the intensity of ambient turbulence. Thin, solid and perforated circular disks have been investigated through high-speed imaging, under laboratory conditions, to reproduce the settling of snow plates and dendrites in quiescent and turbulent flows. Different perforations made it possible to test the parameterisation of the fall speed in quiescent air, based on the geometric description of the solidity of the disk cross-sectional area. Interestingly, different falling styles, ranging from stable horizontal to fluttering and tumbling, were observed to depend significantly on the perforation geometry, which resulted in the stabilisation of the particle rotation and in a modulation of the drag coefficient. Ambient turbulence is observed to primarily induce cross-flow drag on the disks settling in the nonlinear regime, causing a reduction of the settling velocity in all cases investigated. Turbulence also manifests a secondary effect on the disk rotational dynamics, in particular when tumbling and stable falling styles co-exist, based on the phase space defined by the Reynolds number $Re$ and the inertia ratio $I^*$. The interaction between ambient turbulence, particle anisotropy and permeability and the likelihood of tumbling is inferred to be the main reason for the observed settling velocity variability of snow dendrites in nature.

Achieving precise control over the dynamic manipulation of a drop using an external magnetic field may face challenges due to the intricate relationship between the induced magnetisation and the inherent magnetic properties of the drop. Here, we put forward a fundamental theory that elucidates the morphology and behaviour of a ferrofluid droplet immersed in a different, viscous fluid when subjected to a uniform external magnetic field. Unlike previous studies, we introduce an asymptotic model that investigates the dynamic evolution of the drop by examining the local magnetisation as a function of the magnetic field itself. This leads to an additional contribution to the interfacial energy, resulting in an excess normal traction at the interface. Our analytical findings highlight the significant impact of saturation magnetisation and initial susceptibility of the ferrofluid on the resulting dynamic characteristics, which are further explored through comprehensive numerical simulations to address deformations beyond the scope of the asymptotic theory. Supported by benchmark numerical and experimental results, our study suggests that higher magnetic fields and/or greater saturation magnetisation can enhance drop elongation and accelerate its settling process. We develop a regime map illustrating various dynamic events based on the magnetic properties, which could have fundamental implications for the design and control of micro-encapsulations across a wide range of applications, including thermal processing, chemical synthesis, analysis and medical diagnostics.

We employ a novel computational modelling framework to perform high-fidelity direct numerical simulations of aero-structural interactions in bat-inspired membrane wings. The wing of a bat consists of an elastic membrane supported by a highly articulated skeleton, enabling localised control over wing movement and deformation during flight. By modelling these complex deformations, along with realistic wing movements and interactions with the surrounding airflow, we expect to gain new insights into the performance of these unique wings. Our model achieves a high degree of realism by incorporating experimental measurements of the skeleton’s joint movements to guide the fluid–structure interaction simulations. The simulations reveal that different segments of the wing undergo distinct aeroelastic deformations, impacting the flow dynamics and aerodynamic loads. Specifically, the simulations show significant variations in the effectiveness of the wing in generating lift, drag and thrust forces across different segments and regions of the wing. We employ a force partitioning method to analyse the causality of pressure loads over the wing, demonstrating that vortex-induced pressure forces are dominant while added-mass contributions to aerodynamic loads are minimal. This approach also elucidates the role of various flow structures in shaping pressure distributions. Finally, we compare the fully articulated, flexible bat wing with equivalent stiff wings derived from the same kinematics, demonstrating the critical impact of wing articulation and deformation on aerodynamic efficiency.

We consider the drawing of a hollow Newtonian fibre with temperature-dependent viscosity. The drawing is affected by surface tension, inertia, hole pressurisation and externally applied cooling. We apply long-wavelength techniques to determine the steady states and examine their stability. In the presence of surface tension but with no cooling or internal hole pressure, we show the counter-intuitive result that the hole radius at the outlet of the device is a non-monotonic function of the hole radius at the inlet. We also show that if the internal hole is pressurised and the hole size at the inlet is sufficiently large, then the exit temperature of the fibre is a non-monotonic function of the applied cooling rate. We have found a number of surprising mechanisms related to how the various physical effects influence the stability of drawing. For the isothermal case, we show that increasing the internal hole pressure has a destabilising effect for non-zero surface tension while the stability is completely independent of the internal hole pressure for zero surface tension. We further show that there is a complicated interplay between internal hole pressure, external cooling and surface tension in determining the stability and that it is possible that increasing the hole size at the inlet can act to destabilise, then stabilise and finally destabilise the flow. We discuss the mechanisms that determine the counter-intuitive steady-state behaviour and stability.

This study seeks a low-rank representation of turbulent flow data obtained from multiple sources. To uncover such a representation, we consider finding a finite-dimensional manifold that captures underlying turbulent flow structures and characteristics. While nonlinear machine-learning techniques can be considered to seek a low-order manifold from flow field data, there exists an infinite number of transformations between data-driven low-order representations, causing difficulty in understanding turbulent flows on a manifold. Finding a manifold that captures turbulence characteristics becomes further challenging when considering multi-source data together due to the presence of inherent noise or uncertainties and the difference in the spatiotemporal length scale resolved in flow snapshots, which depends on approaches in collecting data. With an example of numerical and experimental data sets of transitional turbulent boundary layers, this study considers an observable-augmented nonlinear autoencoder-based compression, enabling data-driven feature extraction with prior knowledge of turbulence. We show that it is possible to find a low-rank subspace that not only captures structural features of flows across the Reynolds number but also distinguishes the data source. Along with machine-learning-based super-resolution, we further argue that the present manifold can be used to validate the outcome of modern data-driven techniques when training and evaluating across data sets collected through different techniques. The current approach could serve as a foundation for a range of analyses including reduced-complexity modelling and state estimation with multi-source turbulent flow data.

Flow dynamics around a stationary flat plate near a free surface is investigated using time-resolved two-dimensional particle image velocimetry. The study examines variations in angle of attack ($\theta =0^\circ {-}35^\circ {}$), Reynolds number ($Re$ $\approx$ $10^3$$-$3 $\times$ $10^4$) and plate proximity to the free surface ($H^*$). Under symmetric boundary conditions ($H^*\geqslant {15}$), increasing $\theta$ intensifies fluid–plate interaction, resulting in the shedding of leading-edge and trailing-edge vortices (LEV and TEV), each characterised by distinct strengths and sizes. In both symmetric ($H^*\geqslant {15}$) and asymmetric ($H^*=5$) boundary conditions at $\theta \lt 5^\circ {}$, fluid flow follows the contour of the plate, unaffected by Reynolds number. However, at $H^*=5$, three flow regimes emerge: the first Coanda effect (CI), regular shedding (RS) and the second Coanda effect (CII), each influenced by $\theta$ and $Re$. The CI regime dominates at lower angles ($5^\circ {}\leqslant \theta \leqslant 25^\circ {}$) and $Re \leqslant 12\,500$, featuring a Coanda-induced jet-like flow pattern. As the Reynolds number increases, the flow transitions into the RS regime, leading to detachment from the upper surface of the plate. This detachment results in the formation of LEV and TEV in the wake, along with surface deformation, secondary vortices and wavy shear layers beneath the free surface. At $22\,360\lt Re \leqslant 32\,200$ and $5^\circ {} \leqslant \theta \leqslant 25^\circ {}$, in the CII regime, significant surface deformation causes the Coanda effect to reattach the flow to the plate, forming a unique jet-like flow.

The present study investigates the gravity-driven settling dynamics of non-Brownian suspensions consisting of spherical and cubic particles within a triply periodic domain. We numerically examine the impact of solid volume fraction on the evolving microstructure of the suspension using the rigid multiblob method under Stokes flow conditions. Our simulations match macroscopic trends observed in experiments, and align well with established semi-empirical correlations across a broad range of volume fractions. At low to moderate solid volume fractions, the settling mechanism is governed primarily by hydrodynamic interactions between the particles and the surrounding fluid. However, frequent collisions between particles in a highly packed space tend to suppress velocity fluctuations at denser regimes. For dilute suspensions, transport properties are shaped predominantly by an anisotropic microstructure, though this anisotropy diminishes as many-body interactions intensify at higher volume fractions. Notably, cubic particles exhibit lower anisotropy in velocity fluctuations compared to spherical particles, owing to more efficient momentum and energy transfer from the gravity-driven direction to transverse directions. Finally, bidisperse suspensions with mixed particle shapes show enhanced velocity fluctuations, driven by shape-induced variations in drag and increased hydrodynamic disturbances. These fluctuations in turn affect the local sedimentation velocity field, leading to the segregation of particles in the mixture.