When a drop impinges onto a deep liquid pool, it can yield various splashing behaviours, leading to a crown-like structure along the free surface. Under high-speed impact conditions, the upper portion of the thin-walled crown may undergo necking and encapsulate a large bubble, which remains fascinating and is rarely discussed in the literature. In this work, we numerically study this physical process based on the volume-of-fluid and adaptive mesh refinement framework. Our meticulous observations have allowed us to unveil a spectrum of repeatable early-time jet behaviours, vorticity structures and crater evolution, underscoring the rich and complex nature of drop-impact phenomenon. We show that the interplay between aerodynamic pressure and surface tension on the liquid crown could play a significant role in its bending and surface closure. A regime map, incorporating both early-stage jet dynamics and overall bubble-canopy formation, is established across a wide parameter space. This study provides a comprehensive understanding of the diverse splashing regimes, offering insights into the fundamental characteristics of drop-impact phenomenon.

Rotor–stator interactions in turbomachines are characterised by a complex interplay of hydrodynamic instabilities, acoustic pressure waves and receptivity mechanisms, as well as the collision of coherent structures with the blade geometry. An unsteady dual analysis of self-excited instabilities and flow interactions, exemplified by a simple model compressor stage under subsonic conditions, is proposed and presented. Using a low-dissipation sliding-plane implementation, instability-resolving nonlinear-adjoint looping simulations provide detailed sensitivity information that allows for the dissection of the full flow into sub-components linked to distinct flow phenomena. This sensitivity information further links observed flow behaviour to its hydrodynamic or acoustic origin, thereby laying the foundation for a cause-and-effect analysis and for flow control.

Wall-based active and passive flow control for drag reduction in low-Reynolds-number (${\textit{Re}}$) turbulent flows can lead to three typical phenomena: (i) attenuation or (ii) amplification of the near-wall cycle, and (iii) generation of spanwise rollers. The present study conducts direct numerical simulations of a low ${\textit{Re}}$ turbulent channel flow and demonstrates that each flow response can be generated with a wall transpiration at two sets of spatial scales, termed streak and roller scales. The effect of the transpiration is controlled by its relative phase to the background flow, which can be parametrised by the wall pressure. Streak scales (i) attenuate the near-wall cycle if transpiration and wall pressure are approximately in-phase or (ii) amplify it otherwise, and (iii) roller scales energise spanwise rollers when transpiration and wall pressure are out-of-phase. Conditions for establishing these robust phase relations are derived from the analytical solution to the pressure Poisson equation and rely on splitting the pressure into its fast, slow and Stokes component. The importance of each condition depends on the relative magnitude of the pressure components, which is significantly altered by the transpiration. The analogy in flow response suggests that transpiration with the two scale families and their phase relations to the wall pressure represent fundamental building blocks for flows over tailored surfaces including riblets, porous and permeable walls.

The moulting of birds creates different trailing-edge gaps in their wings, which inspires the handling of damaged wings in micro-air vehicles. The effects of the moult gap on aerodynamic performance are investigated by employing a bird-inspired flapping wing model. The aerodynamic performance is evaluated by numerically solving the Navier–Stokes equations for incompressible flows. Moult-gapped wings with different gap widths and positions are compared with the original intact wing in terms of aerodynamic forces and vortex structures. It is found that the decrease in the average lift is slower than that expected from the classical aerodynamic model. The moult gap results in three-dimensional gap vortices, which interact with leading-edge vortices and tip vortices. The interaction generates a pair of parallelly arranged vortex loops on each wing. The downwash momentum associated with this pair of vortex loops is enhanced by the gap vortices. The gap-vortices-enhanced downwash compensates for the loss in the lifting surface, increasing the aerodynamic force per unit area. A composite actuator disk model is proposed based on the vortex loops. The proposed model accounts for not only the finite-span wing effects but also the vortex compensation effects, while the previous quasi-steady model only accounts for the finite-span wing effects.

This study explores interfacial waves in a three-layer fluid system, focusing on the coupling effects between the two interfaces. These effects include resonance induced by inertial coupling and damping caused by viscous coupling. A linear theoretical framework is developed to describe the coupled wave motion and evaluate the impact of interfacial coupling under viscous damping. Additionally, a semi-analytical model is introduced to accurately capture resonance frequency shifts and phase differences due to viscosity. The spiral structure of interfacial waves predicted by the models is confirmed experimentally using the background oriented Schlieren (BOS) method. Further, the model is validated by excellent agreement between theoretical predictions and ultrasonic measurements of wave amplitudes and phase differences. Finally, the study examines mechanical coupling and energy transfer between interfaces under external forcing, elucidating the formation of spiral waves. The accurate treatment of viscous boundary conditions by the semi-analytical model also enables its extension to multilayer fluid systems.

The vertical heated pipe is widely used in thermal engineering applications, as buoyancy can help drive a flow, but several flow regimes are possible: shear-driven turbulence, laminarised flow and convective turbulence. Steady velocity fields that maximise heat transfer have previously been calculated for heated pipe flow, but were calculated independently of buoyancy forces, and hence independently of the flow regime and time-dependent dynamics of the flow. In this work, a variational method is applied to find an optimal body force of limited magnitude $A_0$ that maximises heat transfer for the vertical arrangement, with the velocity field constrained by the full governing equations. In our calculations, mostly at Reynolds number ${\textit{Re}}=3000$, it is found that streamwise-independent rolls remain optimal, as in previous steady optimisations, but that the optimal number of rolls and their radial position is dependent on the flow regime. Surprisingly, while it is generally assumed that turbulence enhances heat transfer, for the strongly forced case, time dependence typically leads to a reduction. Beyond offering potential improvement through the targeting of the roll configuration for this application, wider implications are that optimisations under the steady flow assumption may overestimate improvements in heat transfer, and that strategies that simply aim to induce turbulence may not necessarily be efficient in enhancing heat transfer either. Including time dependence and the full governing equations in the optimisation is challenging but offers further enhancement and improved reliability in prediction.

The inertial migration of hydrogel particles suspended in a Newtonian fluid flowing through a square channel is studied both experimentally and numerically. Experimental results demonstrate significant differences in the focusing positions of the deformable and rigid particles, highlighting the role of particle deformability in inertial migration. At low Reynolds numbers (${Re}$), hydrogel particles migrate towards the centre of the channel cross-section, whereas the rigid spheres exhibit negligible lateral motion. At finite ${Re}$, they focus at four points along the diagonals in the downstream cross-section, in contrast to the rigid particles which focus near the centre of the channel face at similar ${Re}$. Numerical simulations using viscous hyperelastic particles as a model for hydrogel particles reproduced the experimental results for the particle distribution with an appropriate Young’s modulus of the hyperelastic particles. Further numerical simulations over a broader range of ${Re}$ and the capillary number ($Ca$) reveal various focusing patterns of the particles in the channel cross-section. The phase transitions between them are discussed in terms of the inertial lift and the lift due to particle deformation, which would act in the direction towards lower shear. The stability of the channel centre is analysed using an asymptotic expansion approach to the migration force at low ${Re}$ and $Ca$. The theoretical analysis predicts the critical condition for the transition, which is consistent with the direct numerical simulation. These experimental, numerical and theoretical results contribute to a deeper understanding of inertial migration of deformable particles.

Large-scale circulation (LSC) dynamics have been studied in thermal convection driven by heat-releasing particles via the four-way coupled Euler–Lagrange approach. We consider a wide range of Rayleigh–Robert number (${\textit{Rr}}=4.97\times 10^{5} - 4.97 \times 10^{8}$) and density ratio ($\hat {\rho }_r=1- 1000$) that characterize the thermal buoyancy and the particle inertia, respectively. An intriguing flow transition has been found as $\hat {\rho }_r$ continuously increases, involving in sequence three typical LSC regimes, i.e. the bulk-flow-up regime, the marginal regime and the bulk-flow-down (BFD) regime. The comprehensive influence of the LSC regime transition is demonstrated by examining the key flow statistics. As integral flow responses, the heat transfer efficiency and flow intensity change substantially when the LSC regime transition happens, and the thermal boundary layer thicknesses at the top and bottom walls exhibit similar alterations. Significant local accumulation of particles occurs as $\hat {\rho }_r$ increases to a sufficiently high value, resulting in a great modification in the flow dynamics. Specifically, particles aggregate near the sidewalls and heat the local surrounding fluid to generate rising warmer plumes that drive the LSC regime transition. Of interest, well-patterned cellular structures of particles take place near the top wall and obtain notable deviation from the thermal convection cells for the BFD regimes. A mechanical interpretation is proposed and substantiated based on a conceptual vortex–particle model, namely, the centrifugal motion of heat-releasing particles that is confirmed to play a driving role for the LSC regime transition.

A discrete Markov model is proposed to study the interscale dynamics of high Reynolds number wall turbulence. The amplitude modulation of the small turbulent scales due to the interaction with large turbulent scales is investigated for three experimental turbulent boundary layers. Through an appropriate discretisation of the turbulence signals, recently proved universal thermodynamic bounds for discrete-state stochastic systems are shown to apply to continuous-state systems like turbulence, regardless of the distance from the wall and the Reynolds number. Adopting Schnakenberg’s network theory for stochastic processes, we provide evidence for a direct proportionality relation between the mean cycle affinity-based entropy production rate (a stochastic thermodynamic quantity) and a mean entropy production rate associated with the net large-to-small-scale turbulent kinetic energy production. Finally, new insights into the relative arrangement (lag/lead) between large and small scales are provided.

We study the rebound of drops impacting non-wetting substrates at low Weber number ($\textit{We}$) through experiment, direct numerical simulation and reduced-order modelling. Submillimetre-sized drops are normally impacted onto glass slides coated with a thin viscous film that allows them to rebound without contact line formation. Experiments are performed with various drop viscosities, sizes and impact velocities, and we directly measure metrics pertinent to spreading, retraction and rebound using high-speed imaging. We complement experiments with direct numerical simulation and a fully predictive reduced-order model that applies natural geometric and kinematic constraints to simulate the drop shape and dynamics using a spectral method. At low $\textit{We}$, drop rebound is characterised by a weaker dependence of the coefficient of restitution on $\textit{We}$ than in the more commonly studied high-$\textit{We}$ regime, with nearly $\textit{We}$-independent rebound in the inertio-capillary limit, and an increasing contact time as $\textit{We}$ decreases. Drops with higher viscosity or size interact with the substrate longer, have a lower coefficient of restitution and stop bouncing sooner, in good quantitative agreement with our reduced-order model. In the inertio-capillary limit, low-$\textit{We}$ rebound has nearly symmetric spreading and retraction phases and a coefficient of restitution near unity. Increasing $\textit{We}$ or viscosity breaks this symmetry, coinciding with a drop in the coefficient of restitution and an increased dependence on $\textit{We}$. Lastly, the maximum drop deformation and spreading are related through energy arguments, providing a comprehensive framework for drop impact and rebound at low $\textit{We}$.

In this paper, a freely falling circular cylinder attached by a splitter plate in an infinite fluid domain under gravity is investigated numerically. The kinematic modes and wake patterns are summarised, and their parametric sensitivity with the dimensionless plate length ($L^\ast$), the Galileo number ($Ga$) and the cylindric-fluid density ratio ($\rho ^\ast$) is studied. The kinematic modes of a freely falling circular cylinder with a splitter plate can be classified into six types: the steady falling, the steady oblique falling, the small vibration oblique falling, the zigzag oblique falling, the locked falling and the chaotic falling. In the meantime, the wake patterns can be summarised into five types: the steady wake, the 2S wake, the 2P + nS wake, the 2P + 2S wake, and the chaotic wake. The effect of the length of the splitter plate on the vortex shedding characteristics represented by the Strouhal number is also discussed. Further investigation reveals that the attachment of a splitter plate of different lengths to the rear not only influences the kinematic mode and the vortex shedding of the circular cylinder, but also allows the passive and precise control of its falling posture and trajectory. Finally, through theoretical analysis, scaling laws are proposed to estimate the turn angle $\alpha$ and the drift angle $\beta$. The present study can deepen the understanding of similar natural phenomena, such as gliding birds and falling maple seeds, and provide valuable reference for engineering design of drag-reduction devices or air-dropped objects.

We examine the circular, self-similar expansion of frictional rupture due to fluid injected at a constant rate. Fluid migrates within a thin permeable layer parallel to and containing the fault plane. When the Poisson ratio $\nu =0$, self-similarity of the fluid pressure implies fault slip also evolves in an axisymmetric, self-similar manner, reducing the three-dimensional problem for the evolution of fault slip to a single self-similar dimension. The rupture radius grows as $\lambda \sqrt {4\alpha _{hy} t}$, where $t$ is time since the start of injection and $\alpha _{hy}$ is the hydraulic diffusivity of the pore fluid pressure. The prefactor $\lambda$ is determined by a single parameter, $T$, which depends on the pre-injection stress state and injection conditions. The prefactor has the range $0\lt \lambda \lt \infty$, the lower and upper limits of which correspond to marginal pressurisation of the fault and critically stressed conditions, in which the fault-resolved shear stress is close to the pre-injection fault strength. In both limits, we derive solutions for slip by perturbation expansion, to arbitrary order. In the marginally pressurised limit ($\lambda \rightarrow 0$), the perturbation is regular and the series expansion is convergent. For the critically stressed limit ($\lambda \rightarrow \infty$), the perturbation is singular, contains a boundary layer and an outer solution, and the series is divergent. In this case, we provide a composite solution with uniform convergence over the entire rupture using a matched asymptotic expansion. We provide error estimates of the asymptotic expansions in both limits and demonstrate optimal truncation of the singular perturbation in the critically stressed limit.

The flow instabilities in shock-wave–boundary-layer interactions at Mach 6 are comprehensively investigated through compression corner and incident shock cases. The boundary of global stability and the characteristics of globally unstable modes are determined by global stability analysis. In resolvent analysis, cases are categorized into flat plate, no separation, small separation and large separation flows. The optimal response shifts from the first mode in the flat plate case to streaks after the amplification in the interaction region. The amplification of streaks and the first mode (oblique mode) are both attributed to the Görtler instability. Meanwhile, the second mode exhibits minimal growth and higher Mack’s modes appear within the separation bubble. Rounded corner case and linear stability analysis are utilized to further validate the amplification mechanism of the oblique mode.

Motivated by the need for a better understanding of the melting and stability of floating ice bodies, we experimentally investigated the melting of floating ice cylinders. Experiments were carried out in a tank, with ice cylinders with radii between 5 and 12 cm, floating horizontally with their axis perpendicular to gravity. The water in the tank was at room temperature, with salinities ranging from 0 to 35 g l−1. These conditions correspond to Rayleigh numbers in the range 10$^5\lesssim$ Ra $\lesssim$ 10$^9$. The relative density and thus the floating behaviour was varied by employing ice made of H$_2$O–D$_2$O mixtures. In addition, we explored a two-layer stable stratification. We studied the morphological evolution of the cross-section of the cylinders and interpreted our observations in the context of their interaction with the convective flow. The cylinders only capsize in fresh water but not when the ambient is saline. This behaviour can be explained by the balance between the torques exerted by buoyancy and drag, which change as the cylinder melts and rotates. We modelled the oscillatory motion of the cylinders after a capsize as a damped nonlinear oscillator. The downward plume of the ice cylinders follows the expected scalings for a line-source plume. The plume’s Reynolds number scales with Rayleigh number in two regimes, namely Re $\propto$ Ra$^{1/2}$ for Ra $\lt \mathcal{O}(10^7)$ and Re $\propto$ Ra$^{1/3}$ for Ra $\gt \mathcal{O}(10^7)$, and the heat transfer (non-dimensional as Nusselt number) scales as Nu $\propto$ Ra$^{1/3}$. Although the addition of salt substantially alters the solutal, thermal and momentum boundary layers, these scaling relations hold irrespectively of the initial size or the water salinity. While important differences exist between our experiments and real icebergs, our results can qualitatively be connected to natural phenomena occurring in fjords and around isolated icebergs, especially with regard to the melting and capsizing behaviour in stratified waters.

The effects of the external intermittent behaviour on the Kolmogorov constants $C_{k1}$ and $C_2$ in spectral and the physical spaces are investigated using high-resolution direct numerical simulations of a turbulent plane jet. Well-defined $- 5/3$ energy spectrum and $2/3$ structure function can be found in the intermittent flows without large-scale vortex shedding. For different cross-wise positions, the profiles of conditional energy spectra and conditional structure functions exhibit self-similarity at small and intermediate scales when normalised by the conditional Kolmogorov scale of the turbulent region. The conditional Kolmogorov constants are close to those of the fully turbulent flow. The constants $C_{k1}$ and $C_2$ are found to have a power-law dependence on the intermittency factor $\gamma$, that is, $C_{k1}\sim \gamma ^{1/3}$ and $C_{2}\sim \gamma ^{1/3}$, except for the scaling of the structure function in the highly intermittent region with $\gamma =0.25$. In the highly intermittent region, e.g. $\gamma =0.25$, the scaling in the conditional structure function can be considerably influenced by the blocking/sheltering mechanisms of the turbulent/non-turbulent interface (TNTI), leading to slight deviations from self-similarity. We further confirm that the conditional structure function recovers self-similarity after excluding a turbulent region at an average distance of approximately $20$ Kolmogorov length scales from the outer edge of the TNTI, which is comparable to the mean thickness of the TNTI. These findings contribute to the modelling of the edge of a turbulent region.

Wettability quantifies the affinity of a liquid over a substrate and determines whether the surface is repellent or not. When both the liquid and the solid phases are made of the same chemical substance and are at thermal equilibrium, complete wetting is expected in principle, as observed, for instance, with drops of molten metals spreading on their solid counterparts. However, this is not the case for water on ice. Although there is a growing consensus on the partial wetting of water on ice and several estimates available for the value of the associated macroscopic contact angle, the question of whether these values correspond to the contact angle at mechanical and thermal equilibrium is still open. In the present paper, we address this issue experimentally and demonstrate the existence of such a macroscopic contact angle of water on ice, from measurements and theoretical arguments. Indeed, when depositing water droplets on smooth polycrystalline ice layers with accurately controlled surface temperatures, we observe that spreading is unaffected by thermal effects and phase change close enough to the melting point (namely, for undercoolings below 1 K) so that conditions of thermal equilibrium are closely approached. Whereas the short time motion of the contact line is driven by an inertial-capillary balance, the evolution towards mechanical equilibrium is described by a viscous-capillary dynamics and is therefore capillary – and not thermally – related. Moreover, we show that the resulting contact angle remains constant for undercoolings below 1 K. In this way, we show the existence of a non-zero macroscopic contact angle of water on ice under conditions of mechanical and thermal equilibrium, which is very close to $12^\circ$. We anticipate this key finding will significantly improve the understanding of capillary flows in the presence of phase change, which is of special interest in the realm of ice morphogenesis and glaciology, and will also be beneficial with the aim of developing numerical methods for resolving triple-line dynamics.

An addition of polymers can significantly reduce drag in wall-bounded turbulent flows, such as pipes or channels. This phenomenon is accompanied by a noticeable modification of the mean-velocity profile. Starting from the premise that polymers reduce vortex stretching, we derive a theoretical prediction for the mean-velocity profile. After assessing this prediction by numerical experiments of turbulence with reduced vortex stretching, we show that the theory successfully describes experimental measurements of drag reduction in pipe flow.

The turbulent transport of momentum, energy and passive scalar is investigated in the flow around a rectangular cylinder of aspect ratio 5 : 1 – a geometry representative of separating and reattaching flows from sharp-edged bodies. The study is based on direct numerical simulation (DNS) conducted at Reynolds numbers up to ${\textit{Re}} = 14\,000$, based on the cylinder thickness, with Schmidt number fixed at ${\textit{Sc}} = 0.71$. At this Reynolds number, the flow exhibits features of asymptotic high-${\textit{Re}}$ behaviour. Budgets of mean momentum, Reynolds stresses, mean scalar and scalar fluxes provide a detailed view of the underlying transport mechanisms. The mean momentum balance elucidates the role of turbulence in entraining free stream fluid, promoting shear-layer reattachment, sustaining backflow in the recirculation region and regulating wake dynamics through large-scale vortex shedding. The leading-edge shear layer is the main site of turbulence production, with energy injected into streamwise fluctuations and redistributed to cross-flow components by pressure–strain interactions. As ${\textit{Re}}$ increases, vertical fluctuations increasingly return energy to the mean upward flow, stabilising the separation bubble height. Turbulent transport dominates scalar redistribution. Scalar fluxes are primarily generated by interactions between Reynolds stresses and scalar gradient, and modulated by pressure-scalar gradient effects. An a priori evaluation of eddy-viscosity and diffusivity models quantifies the misalignment between modelled and DNS-resolved stress and flux tensors, as well as the inhomogeneity of eddy transport coefficients. This analysis deepens the understanding of transport phenomena in bluff-body flows approaching the asymptotic regime, and underpins the validation and improvement of turbulence models for separating and reattaching flows.

The interaction between deep oceanic currents and an ice base is critical to accurately predict global ice melting rates, yet predictions are often affected by inaccuracies due to inadequate dynamical modelling of the ice–water interface morphology. To improve current predictive models, we numerically investigate the evolution of the ice–water interface under a subsurface turbulent shear-dominated flow, focusing on the time and length scales that govern both global and local morphological features. Based on our previous work (Perissutti, Marchioli & Soldati 2024 Intl J. Multiphase Flow 181, 105007), where we confirmed the existence of a threshold Reynolds number below which only streamwise-oriented topography forms and above which a larger-scale spanwise topography emerges and coexists with the streamwise structures, we explore three orders of magnitude for the Stefan number (the ratio of sensible heat to latent heat). We examine its impact on ice melting and its role in shaping the interface across the two distinct morphodynamic regimes. We identify characteristic time scales of ice melting and demonstrate that the key features of ice morphodynamics scale consistently with the Stefan number and the Péclet number (the ratio of heat advection to diffusion) in both regimes. These scaling relationships can be leveraged to infer the main morphodynamic characteristics of the ice–water interface from direct numerical simulation datasets generated at computationally feasible values of Péclet and Stefan numbers, enabling the incorporation of morphodynamics into geophysical melting models and thereby enhancing their predictive accuracy.

A novel particle-resolved direct numerical simulations (PR-DNS) method for non-spherical particles is developed and validated in the open-source MFiX (Multi-phase Flow with Interphase eXchanges) code for simulating the suspension of non-spherical particles and fluidisation. The model is implemented by coupling superquadric Discrete Element Method-Computational Fluid Dynamics (DEM-CFD) with the immersed boundary method. The model was first validated by applying it to analyse fluid dynamic coefficients ($C_{\!D} , C_{\!L} , C_{\!T}$) of superellipsoids and cylinders at different Reynolds numbers, and the PR-DNS results closely matched those of previous methods, demonstrating the reliability of the current PR-DNS approach. Then, the model was applied to the simulation of the fluidisation of spheres and cylinders. The PR-DNS results were compared with both particle-unresolved superquadric DEM-CFD simulation and experimental data. The pressure drop, height distribution and orientation distribution of particles were analysed. The results show that the PR-DNS method provides a reliable method for reproducing fluidisation experimental results of non-spherical particles. In addition, the comparison of the drag correction coefficients predicted by existing models with that obtained from PR-DNS results indicates the need for a new drag model for particle-unresolved simulation of non-spherical particles.