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.

Using high-fidelity numerical simulations based on a lattice Boltzmann framework, the advection-enhanced transport of a passive scalar from a prolate spheroid in simple shear flow has been thoroughly investigated across various parameters, including the spheroid’s aspect ratio, particle-to-fluid density ratio, Reynolds number (defined as ${\textit{Re}}=\textit{GR}^{2}/\nu$, where $G$ is the flow shear rate, $R$ is the radius of a sphere of the same volume as the spheroid and $\nu$ is the kinematic viscosity of the fluid) and Schmidt number (defined as $\textit{Sc}=\nu /D$, where $D$ is the diffusivity of passive scalar transport). The Reynolds number is constrained to the range of 0 ≤ Re ≤ 1, where the prolate spheroid tumbles around its minor axis, aligned with the vorticity axis, in an equilibrium state. Several key findings have emerged: (i) particle inertia significantly influences the uniformity of the spheroid’s tumbling, affecting flow patterns around the spheroid and, consequently, the modes of scalar transport; (ii) both uniform and non-uniform tumbling generate a scalar line in the fluid with elevated scalar concentration, which sweeps through the wake region and merges with clusters of previously formed scalar lines; (iii) fluid passing over the spheroid carries the passive scalar downstream along these scalar lines; (iv) variations in the uniformity of spheroid tumbling result in distinct flow patterns and scalar transport modes, leading to different transport rates; (v) within the studied parameter ranges, increased particle inertia enhances the scalar transport rate; (vi) when particle inertia is minimal, the dimensionless scalar transport rate for different aspect ratios converges to a common dependence on the Péclet number. These phenomena are analysed in detail.

Interfaces subjected to strong time-periodic horizontal accelerations exhibit striking patterns known as frozen waves. In this study, we experimentally and numerically investigate the formation of such structures in immiscible fluids under high-frequency forcing. In the inertial regime – characterised by large Reynolds and Weber numbers, where viscous and surface tension effects become negligible – we demonstrate that the amplitude of frozen waves scales proportionally with the square of the forcing velocity. These results are consistent with vibro-equilibria theory and extend the theoretical framework proposed by Gréa & Briard (2019 Phys. Rev. Fluids 4, 064608) to immiscible fluids with large density contrasts. Furthermore, we examine the influence of both Reynolds and Weber numbers, not only in the onset of secondary Faraday instabilities – which drive the transition of frozen wave patterns toward a homogenised turbulent state – but also in selecting the dominant wavelength in the final saturated regime.

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.

We numerically investigate the hydrodynamics of an actively heaving flexible foil flapping under a wave surface. The coupled level set and volume-of-fluid method is used to capture the air–water interface, and the immersed-boundary method is used to capture the fluid–structure interaction. A sinusoidal heaving motion is imposed at the foil’s leading edge, and its posterior parts oscillate passively according to its flexible characteristics, allowing dynamic interactions with the wave-induced flow. The propulsive performance of the foil is examined for the influence of three main factors: the ratio of the heaving frequency ($f_{\!f}$) to the wave frequency ($f_w$), the phase difference between the heaving motion and the incident wave ($\mathit \varPhi$) and the submergence depth of the foil ($D$). At $\mathit \varPhi = 0$, the results reveal that the propulsion of the flexible foil benefits from flapping near the wave surface when $f_{\!f}/f_w = 0.5$, and the propulsive efficiency is optimised at $D/L = 1$, where $L$ is the foil’s length. However, when $f_{\!f}/f_w$ = 1.0 and 2.0, the propulsion of the flexible foil is hindered near the wave surface. This hydrodynamic hindrance is closely related to vortex splitting and roll-up phenomena, which induce the formation of a drag wake. By adjusting the phase difference $\mathit \varPhi$, the hindrance in the flexible foil propulsion can be mitigated to enhance propulsive performance. To further understand the relationship between the flapping kinematics and propulsive dynamics, we perform a scaling analysis based on lift force and added mass force, offering good quantification of propulsive performance.

Space–time correlations of velocity and high-Schmidt-number ($Sc \approx 2000$) passive scalar fields are investigated in turbulent pipe flow using particle image velocimetry and planar laser-induced fluorescence, respectively. Both the velocity and scalar fields exhibit characteristic elliptical patterns in their respective space–time correlations. The elliptic approximation model, originally developed for the velocity field, is applied to estimate convection and sweeping velocities for both fields. In both fields, the convection velocity decreases, while the sweeping velocity increases, along the pipe radius. The convection velocity ratio between the scalar and velocity fields shows that high-Schmidt-number scalar fluctuations are advected faster than the velocity fluctuations. Similarly, the sweeping velocity of the scalar fluctuations is found to be larger than that of the velocity fluctuations. Furthermore, the high-Schmidt-number scalar is found to decorrelate more rapidly than the corresponding velocity, with the scalar Taylor microscale distinctly smaller than the velocity Taylor microscale.

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.

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.

Gas-phase turbulence in a bubbling gas–solid fluidised bed is modelled using the data from particle-resolved direct numerical simulations. The subgrid particle-induced turbulent kinetic energy (TKE) is modelled as a function of filter width, filtered solid volume fraction, particle Reynolds number and filtered gas-phase strain rate tensor. Within the volume-filtered framework, we demonstrate that the fluid Reynolds stress models originally developed for a homogeneous system remain applicable to the inhomogeneous fluidised bed, provided that the inhomogeneous drag and particle-induced TKE models are used for the dissipation rate interfacial term. An algebraic model for the anisotropy of gas-phase velocity variance is developed by simplifying the proposed Reynolds stress equation model, which incorporates the effects from both filtered slip velocity and filtered fluid strain rate. The new models are shown to agree well with the direct numerical simulation data of clustered particle settling systems, indicating good applicability of our models for various clustered particle-laden flows.

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.

The locomotion of microorganisms in complex fluids at low Reynolds numbers has been widely studied by ignoring fluid inertia. Here, we combine the asymptotic analysis and numerical simulations to explore the effect of fluid inertia on the dynamic mechanism of microorganisms swimming through viscoelastic fluids using Taylor’s swimming sheet model, undergoing small-amplitude undulations. Surprisingly, fluid inertia can enhance the speed and efficiency of the infinite-length sheet in viscoelastic fluids at finite Reynolds numbers, in stark contrast to the previous results found in Newtonian fluids. Moreover, speed and efficiency slightly exceed those Newtonian values at the small Weissenberg number due to a passive inertial response of the sheet. We associate this with the magnitude of the hydrodynamic force increasing at finite Reynolds numbers. These insights contribute to a deeper understanding of the inertial effect on the locomotion of microorganisms through complex fluids.

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.

We investigate the angular dynamics of a single spheroidal particle with large particle-to-fluid density ratio in simple shear flows, focusing on the influence of the fluid-inertial torque induced by slip velocity. A linear stability analysis is performed to examine how the fluid-inertial torque, viscous shear torque and particle inertia affect the various stable rotation modes, including logrolling, tumbling and aligning modes. As particle inertia increases, bistable or tristable rotation modes emerge depending on initial conditions. For prolate spheroids, three distinct stable-mode regimes are identified, i.e. logrolling, tumbling and tumbling–logrolling (TL). The presence of these modes depends on particle shape and inertia. For oblate spheroids, when the Stokes number is small, we observe monostable modes (logrolling, tumbling and aligning) and bistable modes (TL, aligning–logrolling) varying with different factors. As Stokes number increases, the tristable mode (aligning–tumbling–logrolling) of oblate spheroids appears. These results of the stability analysis further highlight the intricate and significant effect of fluid-inertial torque compared with the results in the absence of fluid-inertial torque. When we apply fluid-inertial torque to the point-particle model, we reproduce the stable rotation modes observed in particle-resolved simulations, which validates the present stability analysis.

An experimental investigation is conducted to examine the tonal noise generation and flow structures of under-expanded jets interacting with a flat plate. The study combines surface pressure, far-field noise and time-resolved Schlieren visualisations to analyse jet dynamics across a range of isentropic Mach numbers (1.1–1.44) and jet-to-plate distances ($H/D$ = 1, 1.5 and 2.5). The results reveal a distinctly non-monotonic relationship between plate height and the amplitude of screech and plate-induced tones. This behaviour is governed by the constructive and destructive interference between the direct acoustic feedback waves of the jet and those reflected from the plate surface. This interference dictates whether the inherent screech mechanism is suppressed or a new plate-induced tone is amplified. Dynamic mode decomposition and wavenumber-spectral analysis reveal that the plate interaction disrupts the balance between downstream-propagating Kelvin–Helmholtz instabilities and upstream-travelling acoustic waves, fundamentally altering the jet’s resonant feedback loops. A key contribution of this work is the establishment of a direct link between flow dynamics and acoustics through advanced statistical analysis. It is shown that the plate installation asymmetrically amplifies the energy of coherent structures within the jet’s lower shear layer. Crucially, the energy content of these dominant shear-layer structures is found to be the primary driver of the far-field tonal noise magnitude. These findings provide a deeper understanding of the complex coupling between flow and acoustics in installed supersonic jets and offer refined guidance for the development of noise mitigation strategies.

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.

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.