In this work we propose a neural operator-based coloured-in-time forcing model to predict space–time characteristics of large-scale turbulent structures in channel flows. The resolvent-based method has emerged as a powerful tool to capture dominant dynamics and associated spatial structures of turbulent flows. However, the method faces the difficulty in modelling the coloured-in-time nonlinear forcing, which often leads to large predictive discrepancies in the frequency spectra of velocity fluctuations. Although the eddy viscosity has been introduced to enhance the resolvent-based method by partially accounting for the forcing colour, it is still not able to accurately capture the decay rate of the time-correlation function. Also, the uncertainty in the modelled eddy viscosity can significantly limit the predictive reliability of the method. In view of these difficulties, we propose using the neural operator based on the DeepONet architecture to model the stochastic forcing as a function of mean velocity and eddy viscosity. Specifically, the DeepONet-based model is constructed to map an arbitrary eddy-viscosity profile and corresponding mean velocity to stochastic forcing spectra based on the direct numerical simulation data at $Re_\tau =180$. Furthermore, the learned forcing model is integrated with the resolvent operator, which enables predicting the space–time flow statistics based on the eddy viscosity and mean velocity from the Reynolds-averaged Navier–Stokes (RANS) method. Our results show that the proposed forcing model can accurately predict the frequency spectra of velocity in channel flows at different characteristic scales. Moreover, the model remains robust across different RANS-provided eddy viscosities and generalises well to $Re_\tau =550$.

The path followed since Faraday’s first observations of acoustic streaming has led to a modern picture of this field as split into separate panels of a tryptic: standing acoustic waves in a channel with uniform background density, known as Rayleigh–Schlichting streaming, with stratified background density, known as baroclinic streaming, and acoustic waves progressing far from the walls under the shape of an attenuated beam, known as Eckart streaming. In their theoretical work, Mushthaq et al. (2025 J. Fluid Mech. 1017, A32) describe in a single continuous parameter space both Rayleigh–Schlichting and baroclinic streaming, thus making a decisive step forward in the frontier between two of these panels. Dealing with a stratification of thermal origin, they identify the level of heating above which baroclinic streaming becomes of the same order of magnitude or greater than Rayleigh–Schlichting streaming. They also depict the major part played by the channel size to wavelength ratio in this problem. This work will be of great help in designing the next generation of experiments concerning acoustic streaming and acoustic management of heat transfer. It is of interest for engineering fields like microfluidics, electronics cooling and biomedical applications. It can also serve as an inspiring basis for academic works in which waves are crossed with stratification.

Direct numerical simulations of two-phase, free-surface flow past a fully submerged, fixed circular cylinder are conducted for transitional Reynolds numbers $400 \leqslant {\textit{Re}} \leqslant 2000$, with Weber number ${\textit{We}} = 1000$, Froude number ${\textit{Fr}} = 1$ and a fixed gap ratio $G = 0.5$. This parameter combination corresponds to the gas entrainment regime characterised by the production of multiscale gas bubbles through interface breakup in the wake, which is of particular interest for its implications in enhancing gas transfer and mixing in environmental and engineering flows, such as air–water gas exchange processes in rivers and oceans, and the design and performance of naval and offshore structures. For ${\textit{Re}}= 400$, the jet forced through the $0.5D$ gap where $D$ is the diameter of the cylinder, efficiently convects opposite-signed vorticity downstream, suppressing the classical von Kármán instability and yielding a quasisteady recirculation bubble. The jet’s stabilising influence, however, breaks down once ${\textit{Re}} \approx 500$: periodic vortex shedding re-emerges and the wake becomes unsteady in spite of the continuing jet. The corresponding dimensionless shedding frequency Strouhal number $St$ grows with ${\textit{Re}}$ as $0.52-72.7{\textit{Re}}^{-1}$. The onset of unsteadiness first shortens the mean separation length but then drives it towards a saturation plateau for higher ${\textit{Re}}$ values. Surface rupture in the turbulent wake fragments entrained air into a multiscale bubble population whose number density follows $S_b(R_{\textit{eff}}) \propto R_{\textit{eff}}^{-6}$, consistent with gravity–capillary breakup in breaking waves, where $R_{\textit{eff}}$ represents the effective radii of the bubbles. Intermittency in entrainment corresponding to vortex shedding contrasts sharply with the finger-like structures observed under laminar conditions, underscoring the role of turbulent mixing. The coupled analysis of vorticity transport, shear-layer instability and bubble statistics elucidates how momentum exchange and air entrainment over a submerged body are governed under non-turbulent and turbulent conditions.

Statistical structure and the underlying energy budget of wall-shear-stress fluctuations are studied in both Poiseuille and Couette flows with emphasis on its streamwise component. Using a dimensional analysis and direct numerical simulation data, it is shown that the spectra of streamwise wall dissipation for $\lambda \lesssim 1000 \delta _\nu$ are asymptotically invariant with the Reynolds number (${\textit{Re}}$), whereas those for $\lambda \gtrsim \delta$ decay with ${\textit{Re}}$ (here, $\lambda$ is a nominal wall-parallel wavelength, and $\delta _\nu$ and $\delta$ are the viscous inner and outer length scales, respectively). The wall dissipation increases with ${\textit{Re}}$ due to the increasing contribution of the spectra at $1000 \delta _\nu \lesssim \lambda \lesssim \delta$. The subsequent analysis of the energy budget shows that the near-wall motions associated with these wall-dissipation spectra are driven mainly by turbulent transport and are ‘inactive’ in the sense that they contain very little Reynolds shear stress (or turbulence production). As such, turbulent-transport spectra near the wall are also found to share the same ${\textit{Re}}$-scaling behaviour with wall dissipation, and this is observed in the spectra of both the wall-normal and inter-scale turbulent transports. The turbulent transport underpinning the increase of wall dissipation with ${\textit{Re}}$ is characterised by energy fluxes towards the wall, together with inverse energy transfer from small to large length scales along the wall-parallel directions.

We highlight the complete transition from liquid-wall-film instability of an annular gas–liquid flow inside a nozzle to spray formation at the trailing edge, aiming to identify two distinct flow regimes of ripple waves and disturbance waves and to clarify their distinct fragmentation mechanisms. Experiments conducted under strictly controlled boundary conditions support our theoretical analysis, revealing that the onset of disturbance waves coincides with the liquid-film Weber number (${\textit{We}}$) of unity, marking a significant change in following fragmentation dynamics. For ${\textit{We}}\lt 0.5$, the liquid wall film forms three-dimensional ripple waves driven by the superposition of Kelvin–Helmholtz and Rayleigh–Taylor (RT) instabilities, with no disturbance waves present. At the trailing edge, the liquid film temporarily accumulates, extends into isolated ligaments along the axial direction via RT instability, and subsequently fragments into droplets through Plateau–Rayleigh instability, displaying a weak coupling between ripple wave dynamics and fragmentation. In contrast, for ${\textit{We}}\gt 0.5$, disturbance waves with long wavelengths and large amplitudes become prominent, superimposed on the base ripple waves. As these disturbance waves reach the trailing edge, they are spontaneously ejected as liquid sheets at the same frequency, forming transverse rims through RT instability and rapidly disintegrating into fine droplets. This regime demonstrates a direct coupling between disturbance-wave dynamics and fragmentation.

This work presents a comprehensive analysis of steady cone-jet electrospray (SCJ-ES) that captures the full range of its steady jet scales within the Taylor-cone electric field. We identify three fundamental regions, each governed by distinct scaling laws and dominant physical mechanisms: (i) the transition region, characterised by the balances that fix the emitted current; (ii) the charge convection-dominated region, where surface charge transport dominates total charge transport and the Taylor field drives jet acceleration; and (iii) the ballistic region, where the jet attains a fixed cylindrical scale before undergoing Rayleigh breakup into charged droplets. This refined theoretical framework harmonises existing models, particularly those using the Taylor–Melcher leaky dielectric model as an electrokinetic approximation for SCJ-ES. Notably, our newly proposed spatial scales achieve a remarkable collapse of published experimental SCJ-ES jet profiles. We also apply this framework to study the charge of resulting droplets using extensive literature data, observing significant differences between weak and strong electrolytes, consistent with recent findings.

We investigate the effectiveness of linear optimal perturbation (LOP) for the flow past a finite span wing in reducing the lifespan of its trailing vortex system. Two approaches, referred to as local and model analysis, are introduced and used for our investigation. Both analyses assume that the baseflow is parallel. Local analysis is suited for intermediate distance from the wing where both tip vortices (TVs) and trailing edge wake (TEW) are present. Its results suggest that the unperturbed baseflow is stable. The separation between TVs and TEW increases downstream and their dynamics appear to be uncoupled at large distance from the wing. When perturbation corresponding to LOP is added to the baseflow, the vortices are displaced forming a helical twist. With time, the maximum displacement initially increases and then saturates. The perturbation retains its compact wavepacket-like structure, and perturbation energy within the tip vortex remains nearly constant. In the model analysis, the far wake is modelled as a pair of counter-rotating $q$-vortices. For low Reynolds number, the flow is stable. However, for higher Reynolds number, the trailing vortices develop Crow instability. Its growth rate is found to be in good agreement with earlier studies. Instability leads to contact of vortices, resulting in the formation of vortex rings. The time for vortex contact decreases with increase in the strength of the initial perturbation. The results suggest that LOP is effective in reducing the lifespan of trailing vortices.

Surfactants are usually added in droplet-based systems to stabilise them. When their concentration exceeds the critical micelle concentration (CMC), they self-assemble into micelles, which act as reservoirs regulating the availability of monomers in the continuous phase, thereby promoting interfacial remobilisation. The monomers get adsorbed onto a drop’s interface to alter its surface tension, and thus, governs how the drop moves within the suspending phase. Indeed, fine tuning droplet trajectories remain crucial in many classical as well as modern applications. Yet, the role of soluble surfactants in modulating droplet movement, especially at high concentrations, hitherto remains poorly understood. To address this, here we investigate the motion and cross-stream migration of a non-deforming drop in an unbounded Poiseuille flow, in the presence of bulk-soluble surfactants at concentrations above the CMC. We build a mixed semi-analytical-cum-numerical framework using spherical harmonics to determine the ensuing velocity and concentration fields. Our results suggest that the drop migrates towards the flow centreline, the extent of which depends on the interplay between the bulk concentration and the sensitivity of the interfacial tension to the surfactant molecules. This propensity for migration plateaus in the presence of micelles, although changing their specific properties seems to have relatively little impact. We further establish that adsorption–desorption between the interface and the bulk tends to suppress migration, while a relatively stronger coupling between bulk and interfacial transport facilitates the same. These findings highlight the crucial role of micelles in droplet motion, with implications in microfluidic control strategies and surfactant-driven flow manipulation.

Flow regimes arising in a T-shaped cell filled with liquid metal under the action of oppositely directed azimuthal electromagnetic forces were investigated. A flow regime map was produced, and the key characteristics of each observed flow type were described. Among the identified flow regimes, funnel rotation, jet flows, tornado-like vortex and their combinations can be distinguished. A flow topologically similar to a magnetohydrodynamic tornado, as well as vertical jet streams, were obtained for the first time without through-flow pumping, using fundamentally planar forces. The study includes experimental observation of flow structures on the free surface of a liquid metal and on the main cell axis, as well as three-dimensional numerical modelling to reconstruct the flow structure in the bulk. A low-melting-point gallium alloy was used as the working fluid. Surface velocity measurements were performed using particle image velocimetry with bubble tracers. Numerical simulations were conducted in a simplified formulation, neglecting free surface deformation.

The evolution mechanisms and suppression strategy of the Richtmyer–Meshkov instability (RMI) at heavy–light interfaces with varying Atwood numbers accelerated by two co-propagating shock waves are investigated through theoretical analysis and experimental evaluation. Existing models describing the complete evolution of once-shocked interfaces and the linear growth of twice-shocked interfaces are examined across low, moderate and high Atwood number regimes, and further refined based on detailed analyses of their limitations. Furthermore, an analytical model for describing the complete evolution of a twice-shocked interface (DS model) is developed through a comprehensive consideration of the shock-compression, start-up, linear and weakly nonlinear evolution processes. The combination of the refined models and DS model enables, for the first time, an accurate prediction of the complete evolution of interfaces subjected to two co-propagating shock waves. Building upon this, the parameter conditions required to manipulate the RMI with varying Atwood numbers are identified. Verification experiments confirm that suppressing the RMI growth at interfaces with various Atwood numbers via a same-side reshock is feasible and predictable. The present study may shed some light on strategies to suppress hydrodynamic instabilities in inertial confinement fusion through integrated adjustment of material densities and shock timings.

We study the temperature–velocity (TV) relation for laminar adiabatic and diabatic hypersonic boundary layers. By applying an asymptotic expansion to the compressible boundary-layer temperature equation, we derive a first-order equation for the TV relation, where the zeroth-order solution is found to be the classical Crocco–Busemann quadratic relation. The ensuing relation predicts accurately the temperature profile by using the velocity for hypersonic boundary layers with Chapman, power and Sutherland viscosity laws, arbitrary heat capacity ratios, variable Prandtl numbers close to unity and Mach number of up to 10. The Mach-number- and wall-temperature-independent quantities in our relation are also investigated. The present relation has the potential to function as a temperature wall model for laminar hypersonic boundary layers, especially for cold-wall cases.

This study employs three-dimensional particle-resolved simulations of planar shocks passing through a suspension of stationary solid particles to study wake-induced gas-phase velocity fluctuations, termed pseudo-turbulence. Strong coupling through interphase momentum and energy exchange generates unsteady wakes and shocklets in the interstitial space between particles. A Helmholtz decomposition of the velocity field shows that the majority of pseudo-turbulence is contained in the solenoidal component from particle wakes, whereas the dilatational component corresponds to the downstream edge of the particle curtain where the flow chokes. One-dimensional phase-averaged statistics of pseudo-turbulent kinetic energy (PTKE) are quantified at various stages of flow development. Reduction in PTKE is observed with increasing shock Mach number due to decreased production, consistent with single-phase compressible turbulence. The anisotropy in Reynolds stresses is found to be relatively constant through the curtain and consistent over all the conditions simulated. Analysis of the budget of PTKE shows that the majority of turbulence is produced through drag and balanced by viscous dissipation. The energy spectra of the streamwise gas-phase velocity fluctuations reveal an inertial subrange that begins at the mean interparticle spacing and decays with a power law of $-5/3$ and steepens to $-3$ at scales much smaller than the particle diameter. A two-equation model is proposed for PTKE and its dissipation. The model is implemented within a hyperbolic Eulerian-based two-fluid model and shows excellent agreement with the particle-resolved simulations.

The interaction between a forward-facing step (FFS) and single-frequency Tollmien–Schlichting (TS) waves is investigated with experiments and two-dimensional (2-D) direct numerical simulations (DNS). Dedicated hot-wire anemometry and particle image velocimetry measurements in the vicinity of the FFS provide characterisation of the perturbation field, as well as validation of the DNS results. Comparison between experiments, 2-D DNS, and linear parabolised stability equations confirm the 2-D nature of the flow and the linearity of the instability mechanisms around the FFS. Upstream of the step, TS waves are gradually amplified by the increasing adverse pressure gradient. In the step vicinity, both mean flow and perturbation field exhibit abrupt distortion, with decoupling of the base flow-oriented growth rate components indicating significant non-modal evolution. Downstream of the step, the mean flow recovers to baseline conditions, but the perturbation field remains highly distorted. Linear stability theory results suggest the presence of superimposed modes on the original TS mode in this region. Despite their decay in the streamwise direction, their presence imprints modifications in the TS wave growth and shape, manifested as the tilting of the perturbation structure in and against the mean flow shear direction. This initiates a reversed Orr mechanism, characterised by a region of stabilisation followed by destabilisation further downstream. Eventually, the TS waves realign to their asymptotic (modal) behaviour. Overall, the FFS destabilises the TS wave far downstream. However, the streamwise extent and magnitude of the stabilisation downstream of the FFS remain significant.

A data assimilation (DA) strategy based on an ensemble Kalman filter (EnKF) is used to enhance the predictive capabilities of scale-resolving numerical tools for the analysis of flows exhibiting cyclic behaviour. More precisely, an ensemble of numerical runs using large-eddy simulations (LES) for a compressible intake flow rig is augmented via the integration of high-fidelity data. This observation is in the form of instantaneous velocity measurements, which are sampled at localised sensors in the physical domain. Two objectives are targeted. The first objective is the calibration of an unsteady inlet condition suitable to capture the cyclic flow investigated. The second objective is the analysis of the synchronisation of the LES velocity field with the available observations. In order to reduce the computational costs required for this analysis, a hyper-localisation procedure (HLEnKF) is proposed and integrated in the library CONES, tailored to perform fast online DA. The proposed strategy performs a satisfactory calibration of the inlet conditions, and its robustness is assessed using two different prior distributions for the free parameters optimised in this task. The DA state estimation is efficient in obtaining accurate local synchronisation of the inferred velocity fields with the observed data. The modal analysis of the kinetic energy field provides additional insight into the improved reconstruction quality of the velocity field. Thus, the HLEnKF shows promising features for the calibration and synchronisation of scale-resolved turbulent flows, opening perspectives of applications for complex phenomena using advanced tools such as digital twins.

We use particle-based simulation to study the rheology of dense suspensions comprising mixtures of small colloids and larger grains subject to contact, lubrication and Brownian forces. These suspensions exhibit shear thinning at low shear rates and shear thickening at high shear rates. By systematically varying the volume fraction of the two species, we demonstrate a monotonic increase in viscosity when grains are added to colloids, but, conversely, a non-monotonic response in both the viscosity and shear-thickening onset when colloids are added to grains. Both effects are most prominent at intermediate shear rates where diffusion and convection play similar roles in the dynamics. We rationalise these results by measuring the maximum flowable volume fraction as functions of the Péclet number and composition, showing that in extreme cases increasing the solids content can disrupt grain contacts and thus allow a jammed suspension to flow. These results establish a constitutive description for the rheology of bidisperse suspensions across the colloidal-to-granular transition, with implications for flow prediction and control in multicomponent particulate systems.

We investigate the inertial migration of slender, axisymmetric, neutrally buoyant filaments in planar Poiseuille flow over a wide range of channel Reynolds numbers (${\textit{Re}}_c \in [0.5, 2000]$). Filaments exhibit complex oscillatory trajectories during tumbling, with the lateral migration velocity strongly coupled to their orientation. Using a singular perturbation approach, we derive a quasi-analytical expression for the migration velocity that captures both instantaneous and period-averaged behaviour. Finite-size effects are incorporated through solid-phase inertia and the influence of fluid inertia on the orientation dynamics. To validate the theory, we develop a fully resolved numerical framework based on the lattice Boltzmann and immersed boundary methods. The theoretical predictions show good agreement with simulation results over a wide range of Reynolds numbers and confinement ratios. Our model outperforms previous theories by providing improved agreement in predicting equilibrium positions across the investigated range of ${\textit{Re}}_c$, particularly at high values. Notably, it captures the inward migration trend toward the channel centreline at high ${\textit{Re}}_c$ and reveals a new dynamics, including the cessation and resumption of tumbling under strong inertial effects. These findings provide a robust foundation for understanding filament migration and guiding inertial microfluidic design.

Double-diffusive linear instability of a power-law fluid flow through porous media with various heat source functions is studied with two permeable infinite parallel walls. The energy balance equation accounts for viscous dissipation, and the temperature and concentration on the boundaries are assumed to be isothermal and isosolutal, respectively. After non-dimensionalisation with appropriate scales, the governing equations are subjected to infinitesimal disturbances on the base flow, and are used to study the stability theory. The results obtained revealed that for large and small values of the Péclet number ($\textit{Pe}$), an increasing source function ($Q_{\textit{Is}}$) delays the onset of convective motion by diminishing the vertical temperature gradient and hence suppressing buoyancy, resulting in a higher critical Rayleigh number (${\textit{Ra}}_c$). In contrast, the non-uniform source ($Q_{\textit{Ns}}$) can destabilise the system due to localised heating, which increases buoyancy and favours the growth of perturbations. Generally, increasing Lewis number (${\textit{Le}}$) tends to suppress the instability under opposing buoyancy conditions, whereas in the case of aiding buoyancy, a sufficiently large throughflow can counteract this stabilising effect. Under the influence of viscous dissipation and source parameters, a pseudo-plastic fluid is more stable compared to a dilatant fluid. In convective rolls, when thermal and solutal diffusivities are equal, dilatant fluids exhibit multicellular convection. Under aiding buoyancy, streamlines develop three counter-rotating vortices, whereas under opposing buoyancy, the pattern attains a symmetric structure.

Wall pressure fluctuations (WPFs) over aerodynamic surfaces contribute to the physical origin of noise generation and vibrational loading. Understanding the generation mechanism of WPFs, especially those exhibiting extremely high amplitudes, is important for advancing design and control in practical applications. In this work, we systematically investigate extreme events of WPFs in turbulent boundary layers and the compressibility effects thereon. The compressibility effects, encompassing extrinsic and intrinsic ones, ranging from weak to strong, are achieved by varying Mach numbers and wall temperatures. A series of datasets at moderate Reynolds numbers obtained from direct numerical simulation are analysed. It is found that the intermittency of WPFs depends weakly on extrinsic compressibility effects, whereas intrinsic compressibility effects significantly enhance intermittency at small scales. Coherent structures related to extreme events are identified using volumetric conditional average. Under extrinsic compressibility effects, extreme events are associated with the weak dilatation structures induced by interactions of high- and low-speed motions. When intrinsic compressibility effects dominate, these events are associated with the strong alternating positive and negative dilatation structures embedded in low-speed streaks. Furthermore, Poisson-equation-based pressure decomposition is performed to partition pressure fluctuations into components governed by distinct physical mechanisms. By analysing the proportion of each pressure component in extreme events, it is found that the contributions of the slow pressure and viscous pressure exhibit weak dependence on the compressibility effects, especially the extrinsic ones, and the varying trend of contributions of the rapid pressure with compressibility effects is opposite to that of the compressible pressure component.

Aerothermal issues in hypersonic transitional swept shock wave/boundary-layer interactions (SBLIs) are critical for the structural safety of high-speed vehicles but remain poorly understood. In this work, previously scarce, high-resolution heat transfer distributions of the hypersonic transitional swept SBLIs, are obtained from fast-responding temperature-sensitive paint (fast TSP) measurements. A series of $34^\circ$ compression ramps with sweep angles ranging from $0^\circ$ to $45^\circ$ are tested in a Mach 12.1 shock tunnel, with a unit Reynolds number of 3.0 $\times$ 10$^{6}$ m$^{-1}$. The fast TSP provides a global view of the three-dimensional aerothermal effects on the ramps, allowing in-depth analysis on the sweep effects and the symmetry of heat transfer. The time-averaged results reveal that the heat flux peak near reattachment shifts upstream with decreasing amplitude as the sweep angle increases, and a second peak emerges in the $45^\circ$ swept ramp due to a type V shock–shock interaction. Downstream of reattachment, the heat flux streaks induced by Görtler-like vortices weaken with increasing sweep angle, whereas their dominant projected wavelengths show little dependence on sweep angle or spanwise location. Away from the ramp’s leading side, the transition onset of the reattached boundary layer gradually approaches the reattachment point. Finally, a general quasi-conical aerothermal symmetry is identified upstream of reattachment, although spanwise variations in transition onset, shock–shock interaction and heat flux streaks are found to disrupt this symmetry downstream of reattachment with varying degrees.