The spatio-temporal evolution of very large-scale coherent structures, also known as superstructures, is investigated in both smooth- and rough-wall boundary layers by means of direct numerical simulations up to a frictional Reynolds number of ${\textit{Re}}_\tau = 3\,150$. One smooth-wall and four rough-wall cases are considered, all developing over a region as long as $\sim$60 times the incoming boundary-layer thickness in the streamwise direction. Bio-inspired, biofouling-type topographies are employed for the rough-wall cases, following the previous work of Womack et al. (2022 J. Fluid Mech. vol. 933, p. A38) and Kaminaris et al. (2023 J. Fluid Mech. vol. 961, p. A23). We utilise three-dimensional time series, as well as multiple two-point correlation functions along the boundary layer to capture the detailed length- and time-scale evolution of the superstructures. The results suggest that the presence of roughness significantly amplifies both the strength and the streamwise growth rate of superstructures. Interestingly, however, their ratios relative to the local boundary-layer thickness, $\mathscr{L}_{\!x}/\delta$ and $\mathscr{L}_z/\delta$, remain constant and independent of the streamwise coordinate, indicating that such scaled length scales might constitute a possible flow invariant. Volumetric correlations revealed that all cases induce structures inclined with respect to the mean-flow direction, with those over the rough-wall topographies exhibiting steeper inclination angles. Finally, via proper orthogonal decomposition, pairs of counter-rotating roll modes were detected and found to flank the high- and low-speed superstructures, supporting the conjecture in the literature regarding the mechanisms responsible for the lateral momentum redistribution. The latter also suggests that the way momentum organises itself in high Reynolds number wall-bounded flows might be independent of the roughness terrain underneath.

We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli’s principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi’s method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.

This study quantitatively investigates the two-dimensional pseudosteady shock refraction at an inclined air–water interface, referred to as the water wedge, in the weak and strong incident shock strength groups. Numerical simulations are employed to validate the predicted refraction sequences from a previous study (Anbu Serene Raj et al. 2024 J. Fluid Mech. 998, A49). A distinctive irregular refraction pattern, referred to as the bound precursor refraction with a Mach reflection, is numerically validated in the weak shock group. Based on the numerical simulations, an enhanced formulation is proposed to determine the sonic line of the incident flow Mach number ($M_b$) in water, thereby providing an appropriate transition condition for an irregular refraction with a Mach reflection to a free precursor refraction with a Mach reflection transition. Furthermore, comparative studies on solid and water wedges of wedge angle $20^\circ$ reveal discernible differences in the shock reflection patterns. The interplay of the energy dissipation due to the transmitted shock wave and the Richtmyer–Meshkov instability at the air–water interface results in the variation of the triple-point trajectory and transition angles between single Mach reflection (SMR) to transitional Mach reflection (TMR) occurring in air.

Solid atmospheric particles, such as ice crystals, pollen, dust, ash and microplastics, strongly influence Earth’s climate, ecosystems and air quality. Previous studies have typically relied on analytical models valid only for very small particles or experiments in liquids, where the particle-to-fluid density ratio $R$ is much lower than values encountered in the atmosphere. We combine a novel experimental set-up with particle-resolved direct numerical simulations to study the settling of sub-millimetre ellipsoids in still air. Particle shapes span elongation and flatness values $ 0.2 \leqslant {\textit{EL}}, {\textit{FL}} \leqslant 1.0$ at a density ratio $ R = 1000$ and particle Reynolds numbers $ 2.1 \lt {\textit{Re}}_{\!p} \lt 4.5$, a regime well below the onset of wake-induced instabilities. Nonetheless, we observe unexpectedly rich dynamics: all non-spherical particles exhibit damped oscillatory motion, and some triaxial ellipsoids follow fully three-dimensional, non-planar trajectories due to rotation about all three axes. Simulations at lower density ratios ($ R = 10, 100$) confirm that these behaviours are driven by strong lateral forces happening only at $R=1000$. Key settling characteristics exhibit nonlinear and non-trivial dependencies on shape. In the two-dimensional phase space of elongation and flatness, settling velocity is symmetric about the principal diagonal ($ {\textit{EL}} = {\textit{FL}}$), while oscillation frequency and damping rate show symmetry about the anti-diagonal. Flatness strongly influences pressure drag, while elongation governs lateral drift and swept volume, which can reach up to ten times the particle diameter and four times the volume-equivalent sphere, respectively.

The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph & Fosdick (1973, Arch. Rat. Mech. Anal. vol. 49, pp. 321–380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Giesekus model. We begin by neglecting inertia to focus on the interaction between gravity, viscoelasticity and surface tension. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface shape on this short time scale, and recover the steady-state shape on a longer time scale. In contrast to the work of Joseph and Fosdick, which used the method of successive approximations to determine the steady shape of the interface, we explicitly derive the interface shape for both steady and transient cases. Subsequently, we reintroduce small but finite inertial effects to investigate their interaction with viscoelasticity, and propose a criterion for determining the conditions under which rod climbing occurs. Through numerical computations, we obtain the transient interface shapes, highlighting the interplay between time-dependent viscoelastic and inertial effects.

Supersonic jets impinging on a ground plane produce a highly unsteady jet shear layer, often resulting in extremely high noise level. The widely accepted mechanism for this jet resonance involves a feedback loop consisting of downstream-travelling coherent structures and upstream-propagating acoustic waves. Despite the importance of coherent structures, often referred to as disturbances, that travel downstream, a comprehensive discussion on the disturbance convection velocity has been limited due to the challenges posed by non-intrusive measurement requirements. To determine the convection velocity of disturbances in the jet shear layer, a high-speed schlieren flow visualisation is carried out, and phase-averaged wave diagrams are constructed from the image sets. The experiments are conducted using a Mach 1.5 jet under various nozzle pressure ratios and across a range of impingement distances. A parametric analysis is performed to examine the influence of nozzle pressure ratio on the convection velocity and phase lead/lag at specific impingement distances. The results reveal that impingement tonal frequency is nearly independent of the disturbance convection velocity, except in cases of staging behaviour. They also demonstrated that slower downstream convection velocity of the disturbance corresponds to larger coherent structures, resulting in increased noise levels. Based on the observation of acoustic standing waves, an acoustic speed-based frequency model has been proposed. With the help of the allowable frequency range calculated from the vortex-sheet model, this model can provide a good approximation for the majority of axisymmetric impingement tonal frequencies.

One-degree-of-freedom flow-induced vibration (FIV) and energy harvesting through FIV of an elastically mounted circular cylinder with mechanically coupled rotation were investigated numerically for low Reynolds number 100, mass ratio 8 and a wide range of reduced velocities. The aims of this study are to investigate the effect of the flow direction angle $\beta$ on the vibration and energy harvesting through FIV. Two types of lock-in are found: vortex-induced vibration (VIV) and galloping. The response amplitude increases with the increase of $\beta$ in both regimes. Both VIV response and galloping regimes are found for $\beta$ = 45° to $\beta$ = 90°. For $\beta$ = −90° to $\beta$ = 0°, only VIV response regimes are found. The fluid force and fluid torque play different roles in exciting/damping the vibration. In the high-amplitude gallop regime, the fluid force excites the vibration, and the torque damps the vibration. Energy harvesting at flow direction angle 90° is investigated as this flow direction has the maximum galloping amplitude. The energy harvesting is achieved by a linear electric damping coefficient in the numerical model. The maximum harvestable power in the galloping regime is significantly greater than that in the VIV regime, and it increases with the increase of the reduced velocity. When the reduced velocity is 20, the harvested power is over 20 times that in the VIV regime, and can further increase if reduced velocity further increases. The maximum efficiency over all simulated parameters is 0.424, occurring when the reduced velocity is 20, and electric damping factor is 0.04.

An arbitrary Lagrangian–Eulerian finite element method and numerical implementation for curved and deforming lipid membranes is presented here. The membrane surface is endowed with a mesh whose in-plane motion need not depend on the in-plane flow of lipids. Instead, in-plane mesh dynamics can be specified arbitrarily. A new class of mesh motions is introduced, where the mesh velocity satisfies the dynamical equations of a user-specified two-dimensional material. A Lagrange multiplier constrains the out-of-plane membrane and mesh velocities to be equal, such that the mesh and material always overlap. An associated numerical inf–sup instability ensues, and is removed by adapting established techniques in the finite element analysis of fluids. In our implementation, the aforementioned Lagrange multiplier is projected onto a discontinuous space of piecewise linear functions. The new mesh motion is compared to established Lagrangian and Eulerian formulations by investigating a pre-eminent numerical benchmark of biological significance: the pulling of a membrane tether from a flat patch and its subsequent lateral translation.

A new statistical definition for the mean turbulent boundary layer (TBL) thickness is introduced, based on identification of the wall-normal location where the streamwise velocity skewness changes sign, from negative to positive, in the outermost region of the boundary layer. Importantly, this definition is independent of arbitrary thresholds, and broadly applicable, including to past single-point measurements. Furthermore, this definition is motivated by the phenomenology of streamwise velocity fluctuations near the turbulent/non-turbulent interface (TNTI), whose local characteristics are shown to be universal for TBLs under low free-stream turbulence conditions (i.e. with or without pressure gradients, surface roughness, etc.) through large-scale experiments, simulations and coherent structure-based modelling. The new approach yields a TBL thickness that is consistent with previous definitions, such as those based on Reynolds shear stress or ‘composite’ mean velocity profiles, and which can be used practically, e.g. to calculate integral thicknesses. Two methods are proposed for estimating the TBL thickness using this definition: one based on simple linear interpolation and the other on fitting a generalised Fourier model to the outer skewness profile. The robustness and limitations of these methods are demonstrated through analysis of several published experimental and numerical datasets, which cover a range of canonical and non-canonical TBLs. These datasets also vary in key characteristics such as wall-normal resolution and measurement noise, particularly in the critical TNTI region.

Floating particles deform the liquid–gas interface, which may lead to capillary repulsion or attraction and aggregation of nearby particles (e.g. the Cheerios effect). Previous studies employed the superposition of capillary multipoles to model interfacial deformation for circular or ellipsoidal particles. However, the induced interfacial deformation depends on the shape of the particle and becomes more complex as the geometric complexity of the particle increases. This study presents a generalised solution for the liquid–gas interface near complex anisotropic particles using the domain perturbations approach. This method enables a closed-form solution for interfacial deformation near particles with an anisotropic shape, as well as the varying height of the pinned liquid–gas contact line. We verified the model via experiments performed with fixed particles held at the water level with shapes such as a circle, hexagon and square, which have either flat or sinusoidal pinned contact lines. Although in this study we concentrate on the equilibrium configuration of the liquid–gas interface in the vicinity of particles placed at fixed positions, our methodology paves the way to explore the interactions among multiple floating anisotropic particles and, thus, the role of particle geometry in self-assembly processes of floating particles.

This research examines in detail the complex nonlinear forces generated when steep waves interact with vertical cylindrical structures, such as those typically used as offshore wind turbine foundations. These interactions, particularly the nonlinear wave forces associated with the secondary load cycle, present unanswered questions about how they are triggered. Our experimental campaigns underscore the occurrence of the secondary load cycle. We also investigate how the vertical distributions of the scattering force, pressure field and wave field affect the nonlinear wave forces associated with the secondary load cycle phenomena. A phase-based harmonic separation method isolates harmonic components of the scattering force’s vertical distribution, pressure field and wave field. This approach facilitates the clear separation of individual harmonics by controlling the phase of incident waves, which offers new insights into the mechanisms of the secondary load cycle. Our findings highlight the importance of complex nonlinear wave–structure interactions in this context. In certain wave regimes, nonlinear forces are locally larger than the linear forces, highlighting the need to consider the secondary load cycle in structural design. In addition, a novel discovery emerges from our comparative analysis, whereby very high-frequency (over the fifth in harmonic and order) oscillations, strongly correlated to wave steepness, have the potential to play a role in structural fatigue. This new in-depth analysis provides a unique insight regarding the complex interplay between severe waves and typical cylindrical offshore structures, adding to our understanding of the secondary load cycle for applications related to offshore wind turbine foundations.

Flows enabled by phoretic mechanisms are of significant interest in several biological and biomedical processes, such as bacterial motion and targeted drug delivery. Here, we develop a homogenisation-based macroscopic boundary condition that describes the effective flow across a diffusio-phoretic microstructured membrane, where the interaction between the membrane walls and the solute particles is modelled via a potential approach. We consider two cases where potential variations occur (i) at the pore scale and (ii) only in the close vicinity of the boundary, allowing for a simplified version of the macroscopic flow description, in the latter case. Chemical interactions at the microscale are rigorously upscaled to macroscopic phoretic solvent velocity and solute flux contributions, and added to the classical permeability and diffusivity properties of the membrane. These properties stem from the solution of Stokes advection–diffusion problems at the microscale, some of them forced by an interaction potential term. Eventually, we show an application of the macroscopic model to develop minimal phoretic pumps, showcasing its suitability for efficient design and optimisation procedures.

Surface quasi-geostrophic (SQG) theory describes the two-dimensional active transport of a scalar field, such as temperature, which – when properly rescaled – shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with fully developed three-dimensional turbulence. In particular, the Kraichnan – Leith – Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling for both scalar and velocity fields, and the presence of intermittency through multifractal scaling was pointed out by Sukhatme & Pierrehumbert (2002 Chaos 12, 439–450), in unforced settings. In this work, we refine the discussion of these statistical analogies, using numerical simulations with up to $16\,384^2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, coupling velocity increments with scalar differences, develop well-defined scaling ranges, highlighting the role of anomalous fluxes of all the scalar moments. However, the clean multiscaling properties of SQG transport are blurred when considering velocity and scalar fields separately. In particular, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires us to resort to generalised notions of scale invariance, such as extended self-similarity and a specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: they evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions that have been repeatedly observed in SQG numerics for the past 20 years.

We study the mechanics of evaporation and precipitate formation in pure and bacteria-laden sessile whole blood droplets in the context of disease diagnostics. Using experimental and theoretical analysis, we show that the evaporation process has three stages based on evaporation rate. In the first stage, edge evaporation results in a gelated contact line along the periphery through a sol–gel phase transition. The intermediate stage consists of a gelated front propagating radially inwards due to capillary flow and droplet height regression in pinned mode, forming a wet-gel phase. We unearthed that the gelation of the entire droplet occurs in the second stage, and the wet-gel formed contains trace amounts of water. In the final slowest stage, the wet gel transforms into a dry gel, leading to desiccation-induced stress forming diverse crack patterns in the precipitate. Slow evaporation in the final stage is quantitatively measured using evaporation of trace water and associated transient delamination of the precipitate. Using the axisymmetric lubrication approximation, we compute the transient droplet height profile and the erythrocytes concentration for the first two stages of evaporation. We show that the precipitate thickness profile computed from the theoretical analysis conforms to the optical profilometry measurements. We show that the drop evaporation rate and final dried residue pattern do not change appreciably within the parameter variation of the bacterial concentration typically found in bacterial infection of living organisms. However, at exceedingly high bacterial concentrations, the cracks formed in the coronal region deviate from the typical radial cracks found in lower concentrations.

This study compares turbulent channel flows over elastic walls with those over rough walls, to explore the role of the dynamic change of shape of the wall in turbulence. The comparison is made meaningful by generating rough walls from instantaneous configurations of elastic cases. The aim of this comparison is to individually understand the role of fluid–structure interaction effects and the role of wall shape/undulations in determining the overall physics of flow near elastic walls. With an increase in the compliance of the wall, qualitatively similar trends for many of the effects produced by a rough wall are also seen in the elastic wall. However, specific features can be observed for the elastic-wall cases only, arising from the mutual interaction between the solid and fluid, leading to a further increase in drag. To understand them, we look at the turbulent structures, which exhibit clear differences across the various configurations: roughness induces only a slight reduction of streamwise coherency, resulting in a situation qualitatively similar to what is found in classical turbulent channel flows, whereas elasticity causes the emergence of a novel dominant spanwise coherency. Additionally, we explored the effect of vertical disturbances on elastic-wall dynamics by comparing with permeable walls having similar (average) wall-normal velocity fluctuations at the interface. The permeable walls were found to have minimal similarities to elastic walls. Overall, we can state that the wall motion caused by the complex fluid–structure interaction contributes significantly to the flow and must be considered when modelling it. In particular, we highlight the emergence of strong wall-normal fluctuations near the wall, which result in strong ejection events, an attribute not observed for rigid walls.

Particle suspensions at the interface of turbulent liquids are governed by the balance of capillary attraction, strain-induced drag and lubrication. Here, we extend previous findings, obtained for small particles whose capillary interactions are dominated by quadrupolar-mode deformation of the interface, to larger spherical and disc-shaped particles experiencing monopole-dominant capillarity. By combining pair-approach experiments, two-dimensional turbulent flow realizations and particle imaging, we demonstrate that particles experiencing monopole-dominant attraction exhibit enhanced clustering compared with their quadrupole-dominant counterparts. We introduce an interaction scale defined by balancing viscous drag and capillary attraction, which is compared with the particle size and interparticle distance. This allows us to map the clustering behaviour onto a parameter space solely defined by those characteristic length scales. This yields a unified framework able to predict the tendency to cluster (and the concentration threshold for those clusters to percolate) in a vast array of fluid–particle systems.

The simulation of turbulent flow requires many degrees of freedom to resolve all the relevant time and length scales. However, due to the dissipative nature of the Navier–Stokes equations, the long-term dynamics is expected to lie on a finite-dimensional invariant manifold with fewer degrees of freedom. In this study, we build low-dimensional data-driven models of pressure-driven flow through a circular pipe. We impose the ‘shift-and-reflect’ symmetry to study the system in a minimal computational cell (e.g. the smallest domain size that sustains turbulence) at a Reynolds number of 2500. We build these models by using autoencoders to parametrise the manifold coordinates and neural ordinary differential equation to describe their time evolution. Direct numerical simulations (DNSs) typically require of the order of $\mathcal{O}(10^5)$ degrees of freedom, while our data-driven framework enables the construction of models with fewer than 20 degrees of freedom. Remarkably, these reduced-order models effectively capture crucial features of the flow, including the streak breakdown. In short-time tracking, these models accurately track the true trajectory for one Lyapunov time, as well as the leading Lyapunov exponent, while at long-times, they successfully capture key aspects of the dynamics such as Reynolds stresses and energy balance. The model can quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. Additionally, we report new exact coherent states found in the DNS with the aid of these low-dimensional models. This approach leads to the discovery of seventeen previously unknown solutions within the turbulent pipe flow system, notably featuring relative periodic orbits characterised by the longest reported periods for such flow conditions.

Turbulence accounts for most of the energy losses associated with the pumping of fluids in pipes. Pulsatile drivings can reduce the drag and energy consumption required to supply a desired mass flux, when compared with steady driving. However, not all pulsation waveforms yield reductions. Here, we compute drag- and energy-optimal driving waveforms using direct numerical simulations and a gradient-free black-box optimisation framework. Specifically, we show that Bayesian optimisation is vastly superior to ordinary gradient-based methods in terms of computational efficiency and robustness, due to its ability to deal with noisy objective functions, as they naturally arise from the finite-time averaging of turbulent flows. We identify optimal waveforms for three Reynolds numbers and two Womersley numbers. At a Reynolds number of $8600$ and a Womersley number of 10, optimal waveforms reduce total energy consumption by 22 % and drag by 37 %. These reductions are rooted in the suppression of turbulence prior to the acceleration phase, the resulting delay in turbulence onset, and the radial localisation of turbulent kinetic energy and production towards the pipe centre. Our results pinpoint that the predominant, steady operation mode of pumping fluids through pipes is far from optimal.

The present experiments investigated the combustion dynamics of single and coaxial laminar diffusion flames within a closed cylindrical acoustic waveguide, focusing on their response to acoustic forcing at a pressure antinode. Nine alternative fuel injectors were used to examine the effect of injector jet diameter and configuration, tube wall thickness, annular-to-inner area and velocity ratio, and jet Reynolds number (below 100) on flame behaviour under different applied frequencies and pressure perturbation amplitudes. Fundamental flame–acoustic coupling phenomena were identified, all of which involved symmetric flame perturbations. These included sustained oscillatory combustion (SOC), multi-frequency periodic liftoff and reattachment (PLOR), permanent flame lift-off (PFLO) with low-level oscillations, and flame blowoff (BO). The phase lag between acoustic forcing and flame response was quantified, providing valuable insights into the coupling dynamics and transition behaviours. Findings revealed how various geometrical and flow characteristics could affect flame stability and resistance to blowoff, even under similar acoustic forcing conditions. Analysis of high-speed spatiotemporal visible imaging using proper orthogonal decomposition (POD) uncovered additional distinct phase portraits and spectral signatures associated with instability transitions, which, coupled with specific dynamical characteristics, enabled new insights into the relevance of injector geometrical characteristics and flow conditions in addressing acoustically coupled combustion instabilities.

The growth of small perturbations in isotropic turbulence is studied using massive ensembles of direct numerical simulations. These ensembles capture the evolution of the ensemble-averaged flow field and the ensemble variance in the fully nonlinear regime of perturbation growth. Evolution equations for these two fields are constructed by applying the ensemble average operator to the Navier–Stokes equations and used to study uncertainty growth in scale and physical space. It is shown that uncertainty growth is described by a flux of energy from the ensemble-averaged flow to the ensemble variance. This flux is formally equivalent to the subgrid scale (SGS) energy fluxes of the turbulence cascade, and can be interpreted as an inverse uncertainty cascade from small to large scales. In the absence of information sources (measurements), the uncertainty cascade is unsteady and leads to the progressive filtering of the small scales in the ensemble-averaged flow, a process that represents the loss of predictability due to chaos. Similar to the kinetic energy cascade, the uncertainty cascade displays an inertial range with a constant average uncertainty flux, which is bounded from below by the average kinetic energy dissipation. Locally in space, uncertainty fluxes differ from the SGS energy fluxes at the same scale, but both have similar statistics and are significantly correlated with each other in space. This suggests that uncertainty propagation is partly connected to the energy cascade and that they share similar mechanisms. These findings open avenues to model uncertainty propagation in turbulence following an approach similar to the SGS models in large-eddy simulations. This is relevant not only to efficiently assess the reliability and accuracy of turbulence forecasts, but also to design uncertainty-robust reconstruction techniques for data assimilation or SGS modelling.