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.

We must address active matter in the context of soft boundaries to bridge the gap between our understanding of active matter and the dynamics of biological systems (represented as active matter) under natural conditions. However, the physics of such active drops (matter) in contact with a soft and deformable surface has remained elusive. In this paper, we attempt to fill this gap and develop a theory for soft, active wetting. Our theory, which accounts for the various free energies for passive substrate and active drops as well as the active stresses, provides an equilibrium description of (active) particle orientation inside the drop and an equilibrium shape of the drop–soft-solid system. We obtain an analytical equation relating the activity to the internal pressure of an active drop. The equilibrium calculation further yields an ordered state of the polarisation field inside the drop. As compared to the non-active drops, the active drops with extensile activity press more into the soft surface, while the active drops with contractile activity either rise out of the soft surface (for smaller magnitude of negative activity) or make the soft surface bulge (for larger magnitude of negative activity). Finally, the three-phase contact line undergoes a rotation that depends on the strength of activity. These findings shed light on the manner in which the active stresses interact with surface tension and elasticity at the fundamental level.

The study of the shape of droplets on surfaces is an important problem in the physics of fluids and has applications in multiple industries, from agrichemical spraying to microfluidic devices. Motivated by these real-world applications, computational predictions for droplet shapes on complex substrates – rough and chemically heterogeneous surfaces – are desired. Grid-based discretisations in axisymmetric coordinates form the basis of well-established numerical solution methods in this area, but when the problem is not axisymmetric, the shape of the contact line and the distribution of the contact angle around it are unknown. Recently, particle methods, such as pairwise-force smoothed particle hydrodynamics (PF-SPH), have been used to conveniently forego explicit enforcement of the contact angle. The pairwise-force model, however, is far from mature, and there is no consensus in the literature on the choice of pairwise-force profile. We propose a new pair of polynomial force profiles with a simple motivation and validate the PF-SPH model in both static and dynamic tests. We demonstrate its capabilities by computing droplet shapes on a physically structured surface, a surface with a hydrophilic stripe and a virtual wheat leaf with both micro-scale roughness and variable wettability. We anticipate that this model can be extended to dynamic scenarios, such as droplet spreading or impaction, in the future.

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 $\textit{We}$.

To study the physics of small-scale properties of homogeneous isotropic turbulence at increasingly high Reynolds numbers, direct numerical simulation results have been obtained for forced isotropic turbulence at Taylor-scale Reynolds number $R_\lambda =2500$ on a $32\,768^3$ three-dimensional periodic domain using a GPU pseudo-spectral code on a 1.1 exaflop GPU supercomputer (Frontier). These simulations employ the multi-resolution independent simulation (MRIS) technique (Yeung & Ravikumar 2020, Phys. Rev. Fluids, vol. 5, 110517) where ensemble averaging is performed over multiple short segments initiated from velocity fields at modest resolution, and subsequently taken to higher resolution in both space and time. Reynolds numbers are increased by reducing the viscosity with the large-scale forcing parameters unchanged. Although MRIS segments at the highest resolution for each Reynolds number last for only a few Kolmogorov time scales, small-scale physics in the dissipation range is well captured – for instance, in the probability density functions and higher moments of the dissipation rate and enstrophy density, which appear to show monotonic trends persisting well beyond the Reynolds number range in prior works in the literature. Attainment of range of length and time scales consistent with classical scaling also reinforces the potential utility of the present high-resolution data for studies of short-time-scale turbulence physics at high Reynolds numbers where full-length simulations spanning many large-eddy time scales are still not accessible. A single snapshot of the $32\,768^3$ data is publicly available for further analyses via the Johns Hopkins Turbulence Database.

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.

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.

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.

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.

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.

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.

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.

Vapour-driven solutal Marangoni effects have been studied extensively due to their potential applications, including mixing, coating, and droplet transport. Recently, the absorption of highly volatile organic liquid molecules into water droplets, which drives Marangoni effects, has gained significant attention due to its intricate and dynamic physical behaviours. To date, steady-state scenarios have been considered mainly by assuming the rapid establishment of vapour–liquid equilibrium. However, recent studies show that the Marangoni flow arises even under uniform vapour concentration, and requires a considerable time to develop fully. It indicates that the vapour–liquid equilibrium takes longer to establish than was previously assumed, despite earlier studies reporting that vapour molecules instantly adsorb on the interface, highlighting the importance of observing transient flow patterns. Here, we experimentally and numerically investigate time-dependent flow structures throughout the entire lifetime of a droplet in ethanol vapour environments. Under two distinct vapour boundary conditions of uniform and localised vapour distributions, a significant flow structure change consistently occurs within the droplet. The time-varying ethanol vapour mass flux from numerical simulation reveals that the flow transition is caused by the high vapour absorption flux at the droplet contact line, due to the geometric singularity there. Based on the detailed analysis of the surface tension gradient along the droplet interface, we identify that the flow transition occurs before and after the vapour–liquid equilibrium is achieved at the droplet contact line, which induces the flow direction change near the contact line.

The presence of salt in seawater significantly affects the melt rate and morphological evolution of ice. This study investigates the melting process of a vertical cylinder in saline water using a combination of laboratory experiments and direct numerical simulations. The two-dimensional (2-D) direct numerical simulations and three-dimensional (3-D) experiments achieve thermal Rayleigh numbers up to $\textit{Ra}_{T}= \mathcal{O} (10^{9} )$ and saline Rayleigh numbers up to $\textit{Ra}_{S}=\mathcal{O} (10^{12} )$. Some 3-D simulations of the vertical ice cylinder are conducted at $\textit{Ra}_{T}= \mathcal{O} (10^{5} )$ to confirm that the results in 2-D simulations are qualitatively similar to those in 3-D simulations. The mean melt rate exhibits a non-monotonic relationship with ambient salinity. With increasing salinity, the mean melt rate initially decreases towards the point where thermal and saline effects balance, after which it increases again. Based on the ambient salinity, the flow can be categorised into three regimes: temperature-driven flow, salinity-driven flow and thermal-saline competing flow. In the temperature-driven and competing flow regimes, we find that the mean melt rate follows a $\textit{Ra}_{T_d}^{1/4}$ scaling, where the subscript $d$ denotes a response parameter. In contrast, in the salinity-driven flow regime, we see a transition from a $\textit{Ra}_{T_d}^{1/4}$ to a $\textit{Ra}_{T_d}^{1/3}$ scaling. Additionally, the mean melt rate follows a $\textit{Ra}_{S_d}^{1/3}$ scaling in this regime. The ice cylinder develops distinct morphologies in different flow regimes. In the thermal-saline competing flow regime, distinctive scallop (dimpled) patterns emerge along the ice cylinder due to the competition between thermal buoyancy and saline buoyancy. We observe these scallop patterns to migrate downwards over time, due to local differences in the melt rate, for which we provide a qualitative explanation.

We introduce a novel approach to derive compressibility corrections for Reynolds-averaged Navier–Stokes (RANS) models. Using this approach, we derive variable-property corrections for wall-bounded flows that take into account the distinct scaling characteristics of the inner and outer layers, extending the earlier work of Otero Rodriguez et al. (Intl J. Heat Fluid Flow, 73, 2018, 114–123). We also propose modifying the eddy viscosity to account for changes in the near-wall damping of turbulence due to intrinsic compressibility effects. The resulting corrections are consistent with our recently proposed velocity transformation (Hasan et al. Phys. Rev. Fluids, 8, 2023, L112601) in the inner layer and the Van Driest velocity transformation in the outer layer. Furthermore, we address some important aspects related to the modelling of the energy equation, primarily focusing on the turbulent Prandtl number and the modelling of the source terms. Compared with the existing state-of-the-art compressibility corrections, the present corrections, combined with accurate modelling of the energy equation, lead to a significant improvement in the results for a wide range of turbulent boundary layers and channel flows. The proposed corrections have the potential to enhance modelling across a range of applications, involving low-speed flows with strong heat transfer, fluids at supercritical pressures, and supersonic and hypersonic flows.

The breakup of viscous liquid threads is governed by a complex interplay of inertial, viscous and capillary stresses. Theoretical predictions near the point of breakup suggest the emergence of a finite-time singularity, leading to universal power laws describing the breakup, characterised by a universal prefactor. Recent stability analyses indicate that, due to the presence of complex eigenvalues, achieving similarity may only be possible through time-damped oscillations, making it unclear when and how self-similar regimes are reached for both visco-inertial and viscous regimes. In this paper, we combine experiments with unprecedented spatio-temporal resolution and highly resolved numerical simulations to investigate the evolution of the liquid free surface during the pinching of a viscous capillary bridge. We experimentally show for the first time that, for viscous fluids the approach to the self-similar solution is composed of a large overshoot of the instantaneous shrinking speed before the system converges to the nonlinear pinch-off similarity solution. In the visco-inertial case, the convergence to the stable solution is oscillatory, whereas in the viscous case, the approach to singularity is monotonic. While our experimental and numerical results are in good agreement in the viscous regime, systematic differences emerge in the visco-inertial regime, potentially because of effects such as polymer polydispersity, which are not incorporated into our numerical model.

Sink flow boundary layers on smooth and rough walls were studied experimentally. In all cases a turbulent, zero-pressure-gradient boundary layer was subject to acceleration with K = 3.2 × 10–6, which suppressed the turbulence in the outer region and produced conditions similar to those in turbulent sink flow cases with lower K. In the smooth-wall case, after the momentum thickness Reynolds number had dropped to about 600, the near-wall turbulence then dropped, resulting in relaminarisation. In the rough-wall cases, the near-wall turbulence was sustained in spite of the strong favourable pressure gradient, and relaminarisation did not occur. A temporary equilibrium appears to occur that is similar to that seen with lower K, in spite of the ratio of the boundary-layer thickness to the roughness height dropping to less than 5. Mean velocity and Reynolds stress profiles, quadrant analysis and turbulence spectra are used to show the development of the boundary layer in response to the pressure gradient and the differences between the rough- and smooth-wall cases. This is believed to be the first study to consider the spatial evolution of constant-K rough-wall boundary layers with K large enough to cause relaminarisation in the smooth-wall case.

Based on the assumption of locally quasi-steady behaviour, Duran & Moreau (2013 J. Fluid Mech. 723, 190–231), assumed that, at a critical nozzle throat, the fluctuations of the Mach number vanish for linear perturbations of a quasi-one-dimensional isentropic flow. This appears to be valid only in the quasi-steady-flow limit. Based on the analytical model of Marble & Candel (1977 J. Sound Vib. 55, 225–243) an alternative boundary condition is obtained, which is valid for nozzle geometries with a finite limit of the second spatial derivative of the cross-section on the subsonic side of the throat. When the nozzle geometry does not satisfy this condition, the application of a quasi-one-dimensional theory becomes questionable. The consequences of this for the quasi-one-dimensional modelling of the acoustic response of choked nozzles are discussed for three specific nozzle geometries. Surprisingly, the relative error in the inlet nozzle admittance and acoustic wave transmission coefficient remains below a per cent, when the quasi-steady boundary condition is used at the throat. However, the prediction of the acoustic fluctuations assuming a quasi-steady critical-throat behaviour is incorrect, because the predicted acoustic field is singular at the throat.

Secondary fragmentation of an impulsively accelerated drop depends on fluid properties and velocity of the ambient flow. The critical Weber number $(\mathit{We}_{cr})$, the minimum Weber number at which a drop undergoes non-vibrational breakup, depends on the density ratio $(\rho )$, the drop $(\mathit{Oh}_d)$ and the ambient $(\mathit{Oh}_o)$ Ohnesorge numbers. The current study uses volume-of-fluid based interface-tracking multiphase flow simulations to quantify the effect of different non-dimensional groups on the threshold at which secondary fragmentation occurs. For $\mathit{Oh}_d \leqslant 0.1$, a decrease in $\mathit{Oh}_d$ was found to significantly influence the breakup morphology, plume formation and $\mathit{We}_{cr}$. The balance between the pressure difference between the poles and the periphery, and the shear stresses on the upstream surface, was found to be controlled by $\rho$ and $\mathit{Oh}_o$. These forces induce flow inside the initially spherical drop, resulting in deformation into pancakes and eventually the breakup morphology of a forward/backward bag. The evolution pathways of the drop morphology based on their non-dimensional groups have been charted. With inclusion of the data from the expanded parameter space, the traditional $\mathit{We}_{cr}-\mathit{Oh}_d$ diagram used to illustrate the dependence of the critical Weber number on $\mathit{Oh}_d$ was found to be inadequate in predicting the minimum initial $\mathit{We}$ required to undergo fragmentation. A new non-dimensional parameter $C_{\textit{breakup}}$ is derived based on the competition between the forces driving the drop deformation and the forces resisting the drop deformation. Tested using available experimental data and current simulations, $C_{\textit{breakup}}$ is found to be a robust predictor for the threshold of drop fragmentation.