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.

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 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 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.

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.

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.

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.

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.

Ice shelves that spread into the ocean can develop rifts that can trigger iceberg calving and enhance ocean-induced melting. Fluid mechanically, this system is analogous to an extensionally dominated radial spreading of a non-Newtonian fluid into a relatively inviscid and denser ambient fluid. Laboratory experiments have shown that rift patterns can emerge when the spreading fluid is shear thinning. Linear stability analysis supports these findings, predicting that while the instability mechanism is active in Newtonian fluids, it is suppressed by stabilising secondary-flow cellular vortices. Here, we explore the fully nonlinear evolution of a radially spreading Newtonian fluid, assessing whether large-amplitude perturbations could drive an instability. We use a quasi-three-dimensional numerical simulation that solves the full nonlinear shallow-shelf approximation, tracing the evolving fluid front, and validate it with known axisymmetric solutions and predictions from linear-stability theory. We find that large-amplitude perturbations induce nonlinear effects that give rise to non-axisymmetric patterns, including cusp-like patterns along the fluid front and complex secondary-flow eddies, which have neither been predicted theoretically nor observed experimentally. However, despite these nonlinear effects, large-amplitude perturbations alone are insufficient to induce rift-like patterns in Newtonian fluids. Strain-rate peaks at the troughs of the fluid front suggest that shear-thinning fluids may become more mobile in these regions, potentially leading to rift formation. This coincides with the likely weakening of stabilising forces as the fluid becomes more shear-thinning. These findings elucidate the critical role of nonlinear viscosity on the formation of rift-like patterns, which is the focus of Part 2 of this study.

We examine how ambient temperature $T$ (23–90 $^\circ \mathrm{C}$) alters the dynamics of spark-induced cavitation bubbles across a range of discharge energies. As $T$ rises, the collapse of an isolated spherical bubble weakens monotonically, as quantified by the Rayleigh collapse factor, minimum volume and maximum collapse velocity. When the bubble is generated near a rigid wall, the same thermal attenuation is reflected in reduced jet speed and diminished migration. Most notably, at $T \gtrsim 70\,^\circ \text{C}$, we observe a previously unreported phenomenon: secondary cavitation nuclei appear adjacent to the primary bubble interface where the local pressure falls below the Blake threshold. The pressure reduction is produced by the over-expansion of the primary bubble itself, not by rarefaction waves as suggested in earlier work. Coalescence between these secondary nuclei and the parent bubble seeds pronounced surface wrinkles that intensify Rayleigh–Taylor instability and promote fission, providing an additional route for collapse strength attenuation. These findings clarify the inception mechanism of high-temperature cavitation and offer physical insight into erosion mitigation in heated liquids.

Plastic pollution in our aquatic systems is a pressing issue, and the spread of these particles is determined by several factors. In this study, the advection and dispersion of negatively buoyant finite-size particles of four different shapes (spheres, circular cylinders, square cylinders and flat cuboids) and two sizes (6 and 9 mm) are investigated in turbulent open-channel flow. The volume, mass and characteristic length are fixed for each size. Four different turbulent conditions are considered, varying the free stream velocity $U_{\infty }=$ 0.25 and 0.38 m s–1 and turbulence intensity ($(u'/U)_\infty =4$ % and 9 %). The particles are released individually from below the water surface. A catch-grid is placed along the bottom floor to mark the particle landing location. The average particle advection distance remains unchanged between the turbulence levels, suggesting that the mean settling velocity is independent of turbulence in this regime. Based on the root mean square of the landing locations, the particle dispersion varies with particle shape, size, settling velocity and turbulent flow conditions. For the square cylinders investigated in this work, the effect of particle shape on dispersion is difficult to predict at low flow velocities and turbulence intensities. As the turbulent fluctuations increase, the dispersion becomes more predictable for all shapes. An empirical expression is proposed to relate turbulent velocity fluctuations, integral length scales, particle settling velocity and particle size to streamwise dispersion. It is found that finite-size inertial particles do not disperse per simple turbulent diffusion, meaning that particle geometry has to be incorporated into dispersion models.

The recirculating flow at the rear of a flat-base three-dimensional body with ground proximity is investigated for different body attitudes defined by the pitch angle varying in the range $-1.5^\circ \lt \alpha \lt +2.6^\circ$ and the yaw angle in the range $0^\circ \lt \beta \lt +12^\circ$. Experiments measuring the three components of the mean velocity field in two perpendicular planes intersecting the recirculation area as well as the base pressure distribution are conducted for 50 different attitudes. They provide a clear correlation between the orientation of the spatially averaged reversed flow and the gradient at the centre of the base pressure distribution. Both vectors are found to be in the same so-called w-plane, that is perpendicular to the base of which the azimuthal position changes with the body attitude due to either the flow orientation at the base separation or sometimes to a ground separation for large nose-up pitch. Numerical simulations of the same geometry realised for 10 attitudes show satisfactory agreement with the force coefficients measured in the experiment. Base flow variations induced by attitude changes are also well captured, particularly that of the w-plane. The full three-dimensional simulation data are used to show that the inner structure of the separation bubble is always a tilted recirculation torus, where the tilt orientation is given by the base pressure gradient. At the bubble closure, a pair of longitudinal vortices symmetrically located on both sides of the w-plane are permanently observed with circulations consistent with the circulation of the dividing streamline separation in the w-plane.

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich’s $f$-plane extension of the Miyata–Maltseva–Choi–Camassa model. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg–de Vries (KdV) equation but with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass–Erdmann corner condition for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation leads to the emergence of strong bursts.

The motionless conducting state of liquid-metal convection with an applied vertical magnetic field confined in a vessel with insulating sidewalls becomes linearly unstable to wall modes through a supercritical pitchfork bifurcation. Nevertheless, we show that the transition proceeds subcritically, with stable finite-amplitude solutions with different symmetries existing at parameter values beneath this linear stability threshold. Under increased thermal driving, the branch born from the linear instability becomes unstable and solutions are attracted to the most subcritical branch, which follows a quasiperiodic route to chaos. Thus, we show that the transition to turbulence is controlled by this subcritical branch and hence turbulent solutions have no connection to the initial linear instability. This is further quantified by observing that the subcritical equilibrium solution sets the spatial symmetry of the turbulent mean flow and thus organises large-scale structures in the turbulent regime.

Asymmetries and anisotropies are widespread in biological systems, including in the structure and dynamics of cilia and eukaryotic flagella. These microscopic, hair-like appendages exhibit asymmetric beating patterns that break time-reversal symmetry needed to facilitate fluid transport at the cellular level. The intrinsic anisotropies in ciliary structure can promote preferential beating directions, further influencing their dynamics. In this study, we employ numerical simulation and bifurcation analysis of a mathematical model of a filament driven by a follower force at its tip to explore how intrinsic curvature and direction-dependent bending stiffnesses impact filament dynamics. Our results show that while intrinsic curvature is indeed able to induce asymmetric beating patterns when filament motion is restricted to a plane, this beating is unstable to out-of-plane perturbations. Furthermore, we find that a three-dimensional whirling state seen for isotropic filament dynamics can be suppressed when sufficient asymmetry or anisotropy are introduced. Finally, for bending stiffness ratios as low as 2, we demonstrate that combining structural anisotropy with intrinsic curvature can stabilise asymmetric beating patterns, highlighting the crucial role of anisotropy in ciliary dynamics.

We study the stability of a steady Eckart streaming jet flowing in a closed cylindrical cavity. This configuration is a generic representation of industrial processes where driving flows in a cavity by means of acoustic forcing offers a contactless way of stirring or controlling flows. Successfully doing so, however, requires sufficient insight into the topology induced by the acoustic beam. This, in turn, raises the more fundamental question of whether the basic jet topology is stable and, when it is not, of the alternative states that end up being acoustically forced. To answer these questions, we consider a flow forced by an axisymmetric diffracting beam of attenuated sound waves emitted by a plane circular transducer at one cavity end. At the opposite end, the jet impingement drives recirculating structures spanning nearly the entire cavity radius. We rely on linear stability analysis (LSA) together with three-dimensional nonlinear simulations to identify the flow destabilisation mechanisms and to determine the bifurcation criticalities. We show that flow destabilisation is closely related to the impingement-driven recirculating structures, and that the ratio $C_R$ between the cavity and the maximum beam radii plays a key role on the flow stability. In total, we identified four mode types destabilising the flow. For $4 \leqslant C_R \leqslant 6$, a non-oscillatory perturbation rooted in the jet impingement triggers a supercritical bifurcation. For $C_R = 3$, the flow destabilises through a subcritical non-oscillatory bifurcation and we explain the topological change of the unstable perturbation by analysing its critical points. Further reducing $C_R$ increases the shear within the flow and gradually moves the instability origin to the shear layer between impingement-induced vortices: for $C_R = 2$, an unstable travelling wave grows out of a subcritical bifurcation, which becomes supercritical for $C_R=1$. For each geometry, the nonlinear three-dimensional (3-D) simulations confirm both the topology and the growth rate of the unstable perturbation returned by LSA. This study offers fundamental insight into the stability of acoustically driven flows in general, but also opens possible pathways to either induce turbulence acoustically or to avoid it in realistic configurations.

Elastoviscoplastic effects on liquid plug propagation and rupture occurring in airways are studied computationally using the Oldroyd-B and Saramito–Herschel–Bulkley models. The relevant parameters are selected from physiological values representative of the eighth-to-tenth generation branches of a typical adult lung. The respiration pushes the liquid plug, depositing a trailing film thicker than the leading film. As a result, the liquid plug gets drained and eventually ruptures. We model airway reopening considering a rigid axisymmetric tube whose inner surface is coated by a thin non-Newtonian liquid film. A critical elastic behaviour is revealed: for low Weissenberg number (subcritical), the viscoelastic stress is released in the liquid plug, while for high Weissenberg number (supercritical), the stretched polymeric chains release their stresses in the trailing film, giving rise to (i) hoop stress that increases the film thickness and (ii) axial stress that leads to a speed-up of the liquid plug. Under supercritical conditions, we also identify a resonance that amplifies the elastic stresses. A mechanical analogy is proposed to elucidate the resonance phenomenon. The occurrence of the resonance is robust upon a variation of Weissenberg number, Laplace number, reference solvent-to-total dynamic viscosity ratio, the surfactant elastoviscoplastic mucus. Our simulations confirm that a presence of surfactants do not significantly affect the results, except for the expected delay of airway reopening due to air–mucus surface contamination. Such a novel elastocapillary mechanism increases the risk of epithelial cell damage regardless of the occurrence of plug rupture.

The inviscid mechanism, driving flow instabilities in a $1:3$, planar and symmetric sudden expansion, is discerned through a sensitivity-based protocol, also referred to as inviscid structural sensitivity analysis, with a specific focus on the onset and nature of the secondary instability. The fundamental idea of this methodology is to change the contribution of viscosity solely in the global stability equations, while freezing the base-flow field at the critical conditions. This is practically implemented by decoupling the Reynolds number that serves as the control parameter for determining the steady base flow from that governing the disturbance evolution, in order to repeat the structural sensitivity analysis while progressively increasing the Reynolds number in the linearised equations only. Accordingly, the sequence of structural sensitivity maps enables us to highlight the flow regions where the inviscid instability mechanism acts. The numerical results reveal that the classical structural sensitivity analysis accurately locates the wavemaker region within the primary recirculation zone, but only its inviscid limit can unveil that the core of the instability coincides with the centre of the primary vortex: a hallmark of an elliptic instability. To validate the global findings, the results of the inviscid structural sensitivity analysis are compared with those obtained from geometric optics. The agreement of the two approaches confirms the inviscid character of the instability, thereby providing a complete picture of the nature of the secondary bifurcation.

We use scanning-tomographic particle image velocimetry introduced by Casey, Sakakibara & Thoroddsen (Phys. Fluids, vol. 25 (2), 2013, p. 025102) to measure the volumetric velocity field in a fully turbulent round jet. The experiments are performed for ${Re}=2640,\, 5280$ and $10\,700.$ Using Fourier-based proper orthogonal decomposition (POD), the dominant modes that describe the velocity and vorticity fields are extracted. We employ a new method of averaging POD modes from different experimental runs using a phase-synchronisation with respect to a common basis. For the dominant azimuthal wavenumber $m=1,$ the first and second POD modes of the axial velocity have opposite signs and appear as embracing helical structures, with opposite twist, while, for the same parameters, POD modes of the radial velocity extend to the axis of symmetry and, interestingly, also show a helical shape. The $(m=1)$-POD modes for the azimuthal vorticity appear as two separate structures, consisting of C-shaped loops in the region away from the axis and helically twisted axial tubes close to the axis. The corresponding axial vorticity modes are cone-like and appear as inclined streaks of alternate sign in the $r$–$z$-plane, similar to velocity streaks seen in wall-bounded shear flows. Temporal analysis of the dynamics shows that a $(m=1)$ two-mode velocity POD representation precesses with a Strouhal number of approximately $St=0.05,$ while the same reconstruction based on vorticity POD modes has a slightly higher Strouhal number of $St=0.06.$