We study stirring and dispersion in steady flow through two-dimensional heterogeneous porous media. A fluid is stirred by the porous medium structure that leads to spatial flow variability and the deformation of fluid elements. While dispersion measures the extension of a solute distribution, stretching underpins solute mixing, chemical reactions and biological activity. Despite this, much less is known about fluid stretching in porous media. Its relation to the dispersion process on the one hand and the flow and medium properties on the other remain open questions. To close this gap, we probe the heterogeneity mechanisms that are at the origin of stretching and dispersion of fluid elements, and quantify the upscaled laws that govern their dynamics. Thus we perform detailed numerical simulations of Darcy flow in two-dimensional porous media characterised by a spatially varying hydraulic conductivity. Dispersion is analysed in terms of first-passage time distributions of fluid elements and the temporal evolution of their displacement mean and variance, while deformation is studied in terms of the probability density function and average elongation of fluid elements. We find that the dispersion and stretching behaviours are closely related. The exponents that govern the decay of the first-passage time distributions and evolution of the displacement variances also determine the evolution of the mean elongation. This dynamics is quantified in a single upscaled theoretical framework that bridges between Eulerian flow structure, large-scale dispersion and stretching of fluid elements.

An innovative air-layer establishment method is developed and experimentally investigated, where superhydrophobic transverse grooves are introduced to modulate localised minimal-flux ventilation. The results demonstrate that, owing to the modulation by the transverse grooves, the characteristics of multiphase flow undergo a fundamental transition from discrete bubbles to a continuous air layer. This modulation is strictly governed by both the free-stream velocity and the groove size, expressed as Reynolds number ($\textit{Re}_D$) and dimensionless groove size ($\delta ^*$), respectively. With a systematic variation in $\textit{Re}_D$ and $\delta ^*$, the macroscopic topology of the air layer evolves through three distinct regimes: (i) continuous air layer (CAL), (ii) partial air layer (PAL) and (iii) bubble shedding (BS). The boundaries for these regimes are delineated within the regime map constructed by $\textit{Re}_D$ and $\delta ^*$. It is established that size effects are negligible for the CAL–PAL transition, whereas the transition from PAL to BS exhibits a pronounced dependence on $\delta ^*$. The two regime transitions therefore follow distinct scaling relations. Neither transition boundary shows a discernible dependence on the solid fraction of the grooves. The dependence of regime transitions on ventilation flux is further examined, and the Weber number ($We_j$) is introduced to characterise the competition of the dynamic pressure of injected gas and the capillary force within grooves. The drag-reduction effectiveness of the groove-modulated air layers was also demonstrated. The present findings elucidate the mechanisms of air-layer establishment and provide a robust framework for the design of superhydrophobic grooved surfaces in air-layer drag-reduction applications.

Linear-model-based estimation provides an equation-based route to reconstruct turbulent fluctuations from limited measurements, but its performance in compressible turbulent boundary layers depends critically on both the inter-layer coherence and the modelling of nonlinear forcing. This study systematically extends and assesses linear-model-based estimation for velocity and temperature fluctuations in compressible turbulent boundary layers, using a logarithmic-region measurement plane at Mach numbers $M=2$, 4 and 6 under both adiabatic- and cold-wall conditions. With the direct numerical simulation databases, the linear coherence spectra are first analysed. Results show that the recoverable information is mainly confined to large scales especially when the estimation plane is located near the wall, whereas the coherent scale range broadens markedly as the estimation location moves towards the measurement plane. The temperature coherence exhibits stronger sensitivity to Mach number and wall thermal condition than the streamwise velocity coherence, especially in cold-wall cases where the non-monotonic mean temperature profile alters the inter-layer thermal correlation. Based on these coherence characteristics, large-scale velocity and temperature snapshots are reconstructed using three levels of linear-model-based estimation. A controlled hierarchy comprising the baseline linear model with white forcing, an eddy-viscosity-corrected operator and a combined eddy-viscosity and forcing model is used to distinguish the effects of operator correction and forcing closure. Finally, a single-plane-calibrated estimation of wall-normal fluctuation-energy profiles is performed using the response cross-spectral density estimated by the Lyapunov formulation. The principal physical insight is that inter-layer coherence determines which scales are recoverable, the eddy-viscosity correction primarily improves the reconstructed response structure and the forcing model modifies the fluctuation amplitude and wall-normal energy redistribution. These roles are distinct in instantaneous and statistical reconstruction and are particularly important for temperature fluctuations under cold-wall conditions.

The dynamics of laterally unconstrained flexible filaments in an inclined uniform flow is investigated numerically, inspired by the self-propelled behaviour of droplets and filaments in a crosswind. It is found that in addition to conventional downwind drifting, the filament can also undergo counterintuitive upwind motion, arising from a supercritical pitchfork bifurcation. Depending on the Reynolds number, bending stiffness and inflow inclination angle, the filament exhibits multiple wake and motion states, including static, quasi-static, vibrational and translational modes. In the reference frame translating laterally with the incoming flow, the upwind and downwind drifting states induced by different initial perturbations form a pair of dynamically equivalent, symmetry-related branches. Force decomposition shows that both the streamwise drag and lateral thrust predominantly originate from the normal force, and the corresponding scaling relations are identified. Stability analysis further reveals that the hydrodynamic forces near the equilibrium states behave as restoring forces. The resulting stable equilibria can be characterised by a generalised hydrodynamic potential, whose potential wells correspond to the stable states of the system. These findings provide insight into the dynamics of flexible structures in uniform flow.

The spectral budget of the vertical heat flux $\langle w\theta \rangle$ is investigated in turbulent Rayleigh–Bénard convection by direct numerical simulation. Eight cases spanning three Rayleigh numbers ($\textit{Ra} = 1\times 10^5$, $6\times 10^6$ and $4\times 10^8$) and up to three geometric aspect ratios each are computed at $\textit{Pr} = 1$. The premultiplied spectra of $\langle w\theta \rangle$ reveal a two-peak structure when both $\textit{Ra}$ and $\varGamma$ are sufficiently large. Production of $\langle w\theta \rangle$ is dominated by temperature fluctuations; production by velocity fluctuations is confined to the near-wall region and the large-scale production that develops at sufficient $\varGamma$ is restricted to the bulk. At large $\varGamma$ an inverse inter-scale cascade transfers $\langle w\theta \rangle$ from intermediate to superstructure scales, the transfer towards the superstructure scale taking place away from the boundary layer. This upscale transfer is specific to the covariance, as the vertical kinetic energy and the temperature variance cascade towards small scales, so that the turbulence selectively transfers the correlation between vertical velocity and temperature towards the superstructure scale, consistent with the strong large-scale coherence of the superstructures. Pressure transport is the only mechanism that delivers large-scale $\langle w\theta \rangle$ towards the wall with appreciable magnitude, yet the pressure–strain term destroys the transported quantity so that the net pressure–temperature interaction is purely negative at all scales. No mechanism in the budget delivers large-scale spectral content into the boundary layer, providing a scale-by-scale explanation for the insensitivity of the Nusselt number to the aspect ratio beyond a moderate threshold.

Sediment transport caused by large earthquakes produces distinctive deposits that help extend our records of these events. To understand the underlying mechanisms of seismically induced sediment transport, we perform particle-resolved simulations of sediment transport over vibrating beds, with earthquake vibrations modelled as sinusoidal horizontal and vertical oscillations. The simulations fully resolve both the carrier flow and the motion of individual grains, and phase-averaged statistics are used to characterise how oscillations modulate turbulence, stress and sediment motion. Analysis of the sediment-transport rate and its phase-averaged decomposition shows a rhythmic response to oscillations, enhanced transport, changes in the thickness of the mobile layer and episodes of bidirectional motion occurring during specific phases of the forcing. Our findings on the influence of oscillation conditions on sediment motions provide mechanistic insight in support of the idea that large earthquakes can induce sediment transport even on flat surfaces.

The stable thermohaline stratification of the ocean gives rise to complex generation, propagation and evolution characteristics of internal wave wakes produced by moving bodies. This study proposes a pycnocline injection approach to construct a stratified flow field representative of typical oceanic pycnoclines and develops a high-precision planar array conductivity measurement system to systematically capture the spatiotemporal evolution of internal wave amplitudes induced by a towed sphere. Using statistical analysis, spectral decomposition and filtering techniques, three distinct internal wave components with different generation mechanisms and propagation behaviours are identified. Their dominant Froude number ranges and spatial distribution characteristics are further clarified. Focusing on the predominant component at high Froude numbers, collapse-stage internal waves, this study examines how key parameters, including waveform, propagation speed, wavelength and wave amplitude, vary with the Froude number. Moreover, by incorporating oscillating-source theory, it is shown that the characteristic semicircular or circular concentric patterns of collapse-stage internal waves can be reasonably explained by pulsating disturbances in the turbulent wake and the resulting vertical restoring motions in the stratified fluid.

The viscoelastic instability of cross-slot flow was first observed experimentally almost half a century ago and reproduced numerically two decades ago, yet its physical origin remains unresolved. We revisit this problem for two-dimensional creeping flow of Oldroyd-B fluid by combining direct numerical simulations, global stability analysis, structural sensitivity analysis and energy-budget analysis. Our simulations reproduce the canonical pitchfork bifurcation, and the stability analysis consistently predicts the threshold and perturbation growth rates. The leading eigenmode consists of a chiral velocity–stress perturbation that tilts and rotates the birefringent strand generated by the extensional flow. Structural sensitivity and energy-budget analyses identify narrow high-extension-rate ridges within the extensional flow as both the spatial core and energetic source of the instability. In these ridges, the stress-based wavemaker colocalizes with large positive disturbance polymeric stress power density, indicating localized transfer of stored elastic energy to the disturbance flow. Analyses of cross-slot variants with rounded corners and with a centred cylinder further reveal that neither sharp corners nor a free central stagnation point is the essential destabilizing ingredient; rather, the instability originates from elastic-energy release in extension-dominated regions characteristic of cross-slot flow.

In this work, we compute the hydrodynamic force and the first and second moments of force acting on a translating spherical droplet immersed in a uniform flow using the reciprocal theorem. We consider the low but finite Reynolds number regime, $\textit{Re} = a U \rho _{\!f} / \mu _{\!f}$, and the dilute limit of small droplet volume fraction $\phi$. Here, $U$ denotes the magnitude of the relative velocity between the phases, $a$ the droplet radius and $\rho _{\!f}$ and $\mu _{\!f}$ the density and viscosity of the continuous phase, respectively. We show that the $O(\textit{Re})$ inertial corrections to the first and second moments of force scale as $O(\rho _{\!f} \phi U^2)$ and $O(a\rho _{\!f}\phi U^2)$, respectively. Equivalently, in dimensionless form, these corrections scale as $O(\phi \textit{Re})$ and $O(({a}/{R})\phi \textit{Re})$, where $R$ denotes the macroscopic length scale. Moreover, the ensemble average of the drag force and the higher-order force moments over the distribution of droplet velocities introduces additional contributions proportional to the velocity variance of the dispersed phase, both in the interphase momentum exchange and in the effective stress of the continuous phase. As a consequence, in dilute emulsions of buoyant droplets, the effective stress depends quadratically on the relative velocity between the phases, on the velocity variance of the dispersed phase and on the spatial gradients of these quantities. Finally, we consider a steady-state pipe flow to examine how the force moments derived in this study affect the behaviour of the two-phase flow in this particular configuration.

When jet noise is cast in the form of a linear input–output problem, the cross-spectral-density (CSD) matrix of the sound field is forced by the CSD of nonlinear interactions via the resolvent operator. The forcing CSD is difficult to compute or measure, but its projection onto the resolvent input space can be identified from the CSD of the sound field. We performed two-point acoustic measurements in isothermal turbulent jets over a broad range of subsonic Mach numbers. The CSD matrices are used to obtain an empirical model of the forcing-projection matrix. We investigate the parametric dependence of this low-rank forcing-projection matrix on Mach number, frequency and azimuthal mode, and we propose an empirical model that captures downstream radiation with 2 dB precision for subsonic jets.

We examine the onset of wave breaking in highly viscous gravity-driven vertical falling liquid films in a quiescent atmosphere via multiphase direct numerical simulations. Both Newtonian and Herschel–Bulkley liquids are considered. Analysis of the force balance acting on wave humps shows that the transition from travelling waves to breaking states is governed primarily by the competition between gravitational and capillary forces. The transition is primarily characterised by a criterion based on the Eötvös number (quantifying gravitational to capillary forces), which we identify through a comprehensive parameter study involving over 500 simulations, where we vary the Reynolds number, the plasticity number and the power-law index. We provide a regression model for $\textit{Eo}_{\textit{crit}}$ in terms of these parameters, as well as for the corresponding wetting rate. The parameter study indicates that breaking generally occurs for large film thicknesses and high viscosities, yielding large Eötvös numbers while remaining in the wavy–laminar flow regime. Close to the onset of breaking, we observe overhanging travelling wave solutions, where the wave front deforms into a steady overhang, and this state generally precedes wave breaking irrespective of the investigated rheologies. Our results establish a concise criterion for predicting wave breaking in falling films involving both Newtonian and complex fluids representative of different industrial processes, and have implications for heat and mass transfer, bubble entrainment and liquid residence time.

We report results from experiments and direct numerical simulation (DNS) of the temperature and velocity fields in horizontal convection (HC) over a range of Rayleigh numbers $10^{8} \leqslant Ra \leqslant 5\times 10^{12}$ and Prandtl numbers $4.3 \leqslant Pr \leqslant 7$ in a rectangular cell of aspect ratio $\varGamma \equiv L:W:H = 10:1:1$ ($L$, $W$ and $H$ denote the length, width and height of the cell, respectively). The DNS considers four combinations of velocity and thermal boundary conditions (BCs), spanning free-slip and no-slip velocity constraints together with isothermal and adiabatic thermal conditions. In the experiments, the upper boundary consists of an open water–air interface, which approximates a slippery surface. The DNS results reveal systematic differences in the thermal boundary-layer thickness, vertical temperature gradients and overall flow organisation across the four BC configurations. Building on these findings, we develop a theoretical model based on a prescribed streamfunction that satisfies the relevant boundary constraints. From this formulation, general analytical expressions for the vertical mean temperature profile are derived for each configuration, capturing the interplay among advection, diffusion and the imposed BCs. The model successfully reproduces self-similar features of the temperature profiles observed in the DNS and agrees with available experimental data. This framework thus provides a unified description of vertical temperature and velocity structure in HC and clarifies how velocity and thermal BCs jointly regulate boundary-layer formation and heat transport efficiency. The findings also offer insight into analogous geophysical processes, including urban near-surface heating and oceanic stratification.

Air lubrication regimes were studied using simultaneous drag force measurements and multi-plane imaging to characterise the regimes and identify the governing mechanisms of drag reduction. Bubbly, transitional and air layer regimes are identified over a large range of free-stream velocities ($U_{\infty }$), air flow rates ($Q_{\textit{air}}$) and Froude-depth numbers ($\textit{Fr}_d$). For the lowest $U_{\infty }$, drag reduction lags significantly behind the non-wetted area coverage in all cases and no simple correlation exists. Within the bubbly regime, a drag increase is found for low $U_{\infty }$ with large, slow-moving bubbles forming a single layer over the plate height. For higher velocities, bubbles become smaller and disperse vertically, while the drag starts decreasing. For higher $Q_{\textit{air}}$, irrespective of $U_{\infty }$, air patches start to form (transitional regime) and drag monotonically decreases, with the onset of the air layer regime at 60 % drag reduction. A new scaling of the associated critical $Q_{\textit{air}}$ is proposed, combining the air exit velocity, the liquid velocity close to the air layer and $\textit{Fr}_d$. For a further increase of $Q_{\textit{air}}$ and a low $U_{\infty }$, a thicker and smoother air layer is formed with even lower drag; for higher $U_{\infty }$, marginal differences are observed. The air layer morphology is significantly altered, however, depending on $\textit{Fr}_d$: for $\textit{Fr}_d\gt 0.7$, it is unbounded, extending beyond the current test section length, and for subcritical conditions (deep water regime, $\textit{Fr}_d\lt 0.61$) a closure is formed and the air layer transitions to a cavity of a specific length.

We investigate head-on droplet–bubble collisions in a quiescent, surfactant-free viscous liquid using three-dimensional volume-of-fluid simulations. By systematically varying the bubble-to-droplet radius ratio ($0.5\le \varDelta \le 2$) and Weber number ($3\le \textit{We}\le 270$) at small Ohnesorge number ($ \textit{Oh}\lt 1$), we identify the mechanisms governing interfacial deformation, attachment, jet formation and vortex evolution during an isolated collision. Four distinct collision regimes emerge, ranging from oscillatory attachment to jet-mediated penetration and coupled toroidal deformations. A transition from weak- to inertia-dominated dynamics occurs at a critical Weber number of $ \textit{We}\approx 23$, nearly independent of $\varDelta$, whereas the onset of complete jet penetration depends strongly on the size ratio. Comparison with the unsteady Oseen solution for an impulsively accelerated rigid sphere establishes a hydrodynamic baseline that isolates the effects of interface mobility and deformation, revealing the progressive emergence of toroidal vortices, interfacial oscillations and jetting. An energy analysis demonstrates that jet formation is controlled by the conversion of impact kinetic energy into surface energy, yielding a characteristic jet velocity comparable to the impact speed, while viscous dissipation suppresses jetting above a critical Ohnesorge number scaling as $ \textit{Oh}_c\sim We^{-1/2}$. These results provide a unified physical framework linking inertia, capillarity and vortex dynamics in elementary droplet–bubble interactions, offering predictive insight into multiphase processes such as flotation, aeration, emulsification and liquid–liquid separation.

Three-dimensional experiments are presented on a school of three pitching hydrofoils. Two side-by-side leader foils maintain the same relative positions while the location of a third follower foil is varied. Force and flow measurements detail the mechanisms leading to collective thrust and efficiency 58 % and 24 % higher than isolated foils, respectively. In traditional drafting, a follower is positioned in the wake of an upstream object. In wakes with a net momentum deficit, drafting reduces drag by lowering oncoming flow speed. By contrast, wakes from oscillatory swimmers feature strong momentum surplus regions, which increases drag by increasing oncoming flow. Despite that, our results show that the best performance benefits occur for compact schools where the follower is directly in the vortex wake of a leader, whereas regions of reduced mean flow do not improve performance. The performance benefits are driven by vortex–body interactions that increase the thrust and efficiency of the follower and by body-to-body upstream interactions that reduce the power of the leaders. There is an optimal spatial phase to maximise the thrust and efficiency of the follower that depends upon the actual wake wavelength rather than the estimated wavelength used in previous literature. Moreover, wake breakdown, and its associated elimination of vortex–body performance benefits, is not observed within at least three chord lengths downstream of the leaders. Lastly, cross-stream stability measurements of the downstream foil indicate that compact, high-performance formations may require active control strategies in order to maintain their organisation and maximise the hydrodynamic benefits of schooling.

The onset and control of unsteadiness in an idealised porous medium are investigated using global stability and sensitivity analysis. The flow is modelled by a periodic representative elementary volume composed of tandem-arranged circular cylinders, and the Hopf bifurcation is characterised over a broad porosity range ($0.45 \leqslant \phi \leqslant {0.95}$). The neutral stability curve exhibits an approximately linear dependence on $\phi$, indicating that porosity serves as an effective predictor for the transition threshold. In the vicinity of instability onset, several eigenmode families emerge, clarifying the organisation of the competing branches. Energy budget analysis further shows that perturbation production is dominated by the lift-up mechanism. In addition, adjoint-based sensitivity analysis near criticality identifies where a localised steady forcing produces the largest drift of the leading global mode. The result highlights the throat of the main passage as the region of strongest receptivity, offering a clear target and an efficient tool for designing flow-control strategies.

Bioinspired propulsion demonstrates broad potential for engineering applications. In this study, the hydrodynamic behaviour of an undulating hydrofoil operating at a fixed submergence depth near the free surface is numerically investigated using the smoothed particle hydrodynamics method. The body-wave wavelength $\lambda$ is introduced to characterise the spatial pattern of the prescribed undulatory motion of the hydrofoil, and the effects of submergence depth and wavelength on propulsion performance are systematically analysed together with flow field structures. The results show that cruising speed decreases monotonically with decreasing submergence depth, whereas propulsion efficiency exhibits a non-monotonic trend, indicating an optimal submergence depth for maximum efficiency. Propulsion efficiency decreases monotonically with increasing wavelength, whereas, at shallow submergence depths, an optimal wavelength exists that maximises the cruising speed. When operating close to the free surface, the hydrofoil experiences an attractive force towards the surface; therefore, a negative angle of attack is required to maintain a fixed depth. Free-surface wave generation intensifies with decreasing submergence depth and increasing wavelength; however, under mild wave-breaking conditions, the cruising efficiency of near-surface hydrofoils may still exceed that achieved in deep-water operation. In addition, a scaling relationship between cruising speed, submergence depth and wavelength is obtained through curve fitting and can be used to predict cruising speed. These results provide useful guidance for the optimisation and control of undulating hydrofoils operating near the free surface.

The intrinsic effect of shear-thinning on successive instabilities in narrow-gap Taylor–Couette flow is investigated using direct numerical simulation (DNS) with axially periodic boundary conditions, thereby eliminating endwall effects. The Carreau model is employed with three shear-thinning indices ($n = 0.52$, $0.62$, $0.68$), matching the experimental parameters of Cagney and Balabani (Phys. Fluids, 2019, vol. 31, p. 053102). Shear-thinning exerts opposing influences on the two instabilities. For the primary transition from circular Couette flow to Taylor vortex flow, enhanced shear-thinning (decreasing $n$) is destabilising, with the critical Reynolds number $\textit{Re}_{c,1}$ decreasing from $101$ to $94$. In contrast, the secondary transition to wavy vortex flow is strongly stabilised, with $\textit{Re}_{c,2}$ increasing from $115$ to $142$ over the same range of $n$. This opposing behaviour arises from a competition between two consequences of radial viscosity variation: the reduced viscosity near the inner wall promotes centrifugal disturbances (destabilising), while the elevated viscosity in the bulk gap damps inter-vortex oscillations (stabilising). Comparison with the non-monotonic experimental measurements of Cagney and Balabani (Phys. Fluids, 2019, vol. 31, p. 053102) shows that the observed discrepancies originate from endwall-induced wavelength constraints and vortex drift rather than intrinsic rheological effects. These results establish the intrinsic bifurcation structure of shear-thinning Taylor–Couette flow in the narrow-gap configuration and clarify the distinct roles of rheology and confinement in successive flow transitions.

In this study, direct numerical simulations are performed to investigate inertial particle transport in supersonic turbulent flow in a straight-to-bent pipe. Three particle populations with viscous Stokes numbers (St+) of 0.9, 4 and 16 are considered. It is revealed that under the combined influence of wall confinement, flow compressibility and centrifugal forces, distinct turbulent structures emerge in different pipe regions, leading to inertia-dependent particle dynamics. In the straight section, particle streaks and near-wall accumulation are more significant for St+ = 4 particles. When entering the curved section, Görtler vortices along the outer bend enhance the streak formation and the accumulation of St+ = 16 particles. Further downstream, expansion waves suppress Görtler vortices while Dean vortices become dominant, transporting particles along the sidewalls towards the inner bend, thereby reducing outer-wall collisions and accumulation. Particles tend to escape the confinement of Dean vortices and accumulate on the sidewalls. A larger Stokes number results in an earlier detachment, accompanied by higher collision frequency, particle accumulation and slip velocity. In contrast, small inertia particles follow the vortical motions more closely, migrating into the inner low-speed region and resulting in a higher local particle concentration. These different behaviours stem primarily from differences in the centripetal force imparted by the flow in the streamwise and circumferential directions. The strongest centrifugal effect occurs in the areas on both lateral sides of the pipeline. These elucidations offer new physical insights for modelling and predicting particle-laden turbulent flows in curved geometries.