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.

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.

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.

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.

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.

The present work brings to light the vibrations emerging when a circular cylinder, elastically mounted along a rectilinear path in quiescent fluid, is subjected to a forced rotation about its axis. These rotation-induced vibrations (RIV) are explored numerically for ranges of the four governing parameters. The Reynolds number and the reduced velocity (inverse of the non-dimensional natural frequency of the oscillator), based on the surface velocity of the rotating body and its diameter, are varied up to $100$ and $250$, respectively, and the structural damping ratio up to $50\,\%$. The structure to displaced fluid mass ratio ranges from $0.1$ to $1000$. Vibrations are found to occur over a vast region of the parameter space, including the four orders of magnitude of the mass ratio under study, and high levels of structural damping. The amplitude of RIV may exceed $30$ body diameters, while their frequency varies and deviates from the oscillator natural frequency, even though it is always lower. Despite its simplicity and the steady nature of the actuation, the system exhibits a considerable diversity of behaviours. Three distinct RIV regimes are encountered: two periodic regimes whose responses differ by their spectral contents, i.e. sinusoidal versus multi-harmonic, and an aperiodic regime. These regimes are all closely connected to flow unsteadiness, in particular via the interplay of the cylinder with previously formed vortices, which persist in the vicinity of the body.

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 $\textit{We}$.

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.

In this paper we propose a novel control strategy for modulating nonlinear flapping and symmetry-breaking (S-B) bifurcations of a piezoelectric metamaterial beam behind a circular cylinder subjected to viscous flow. The beam incorporates distributed piezoelectric meta-cells connected via unidirectional circuits to enable self-sensing and adaptive control. A strongly coupled nonlinear fluid-structure-electro-control model within an arbitrary Lagrangian–Eulerian framework is developed for predicting the flapping dynamics of the large deformable piezoelectric metamaterial beam. The system exhibits multiple flow-induced modes, including limit-cycle oscillations, subharmonic responses and S-B deflections. These dynamic regimes arise from nonlinear bifurcations of the system, namely the period-doubling and spontaneous S-B bifurcations. Flapping control and wake topology transition of the system is achieved by suppressing the periodic-doubling bifurcation based on the vibration rebound effect through a self-sensing and adaptive-actuation mechanism of the beam. Floquet stability analysis confirms the effectiveness of control in delaying instability onset and suppressing chaotic transitions. Symmetry modulation of the beam is achieved via the localised perturbations induced from the piezoelectric meta-cells, which reshape the stability of the system. The transition from S-B mode to symmetry-recovery mode reflects a shift from a flow-separation-dominated to vibration-dominated vortex shedding pattern. This symmetry transition reorganises the energy exchange pathways between the flow and the beam. Quantitative analyses of the wake recovery and the energy harvesting efficiency confirm enhanced flow energy conversion under control. These results establish a framework for bifurcation control of slender structures in viscous flow, providing potential applications for underwater energy harvesting and flexible propulsion in unsteady environments.

Equilibrium, travelling-wave and periodic-orbit solutions of the Navier–Stokes equations provide a promising avenue for investigating the structure, dynamics and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille and pipe flow. However, the organisation of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organisation and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new travelling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one travelling-wave solution whose symmetry involves shifts by one third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behaviour of shear flows.

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.

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.

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.

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.