The coherent structures in supersonic turbulent boundary layers, particularly wall-attached Reynolds stress structures (RSS), exhibit significant correlations with skin-friction drag generation. However, the intrinsic coupling between streamwise motions, wall-normal motions, and evolutionary processes of RSS has obscured the precise mechanisms governing their scale-dependent interactions. This study investigates the generation mechanisms of high/low skin friction in a supersonic turbulent boundary layer via direct numerical simulations. A novel methodology combining a time-resolved clustering method and conditionally averaged skin-friction decomposition is employed, focusing on wall-attached RSS (Q2 ejections and Q4 sweeps) of varying wall-normal scales ($l_y^+$), which decomposes the skin-friction coefficient into contributions from streamwise ($C_{\kern-2.5pt f}^{M_x}$), wall-normal ($C_{\kern-2.5pt f}^{M_y}$) and spanwise ($C_{\kern-2.5pt f}^{M_z}$) motions, etc. Results show that $C_{\kern-2.5pt f}^{M_y}$ dominates friction generation, while $C_{\kern-2.5pt f}^{M_z}$ attenuates it. A significant scale dependence is revealed: $C_{\kern-2.5pt f}^{M_x}$ counteracts friction for large-scale structures, contrary to its effects at smaller scales. These three terms reflect the influence of momentum transport by RSS on skin friction, which is specifically manifested in the convective acceleration. The unsteady term ($C_{\kern-2.5pt f}^U$) partially offsets $C_{\kern-2.5pt f}^{M_x}$, linked to structural evolution phases: growth dominates for small scales, and decay for large scales. These findings elucidate the scale-dependent momentum transport mechanisms governing skin friction, providing insights for drag-reduction strategies in high-speed flows.

The transition between dripping and jetting regimes of capillary jets is crucial for applications such as ink-jet printing and drug release. A liquid jet emitting from a nozzle usually exhibits a non-uniform initial velocity profile, influencing the transition between regimes, with the role of velocity relaxation remaining largely unexplored. Here we investigate the dripping–jetting transition of a capillary jet with velocity relaxation through a combination of experiments and stability analysis. Experimental measurements show that velocity relaxation consistently lowers the critical transition Weber number, ${\textit{We}}_c$, contradicting predictions from the classic local spatio-temporal stability theory. To resolve this discrepancy, we developed a global stability model accounting for the velocity relaxation to calculate ${\textit{We}}_c$ at different Reynolds numbers (${\textit{Re}}$). Our model provides accurate predictions for jets both with and without velocity relaxation and reveals that the key to the discrepancy lies primarily in the non-parallel effects of jet disturbances caused by velocity relaxation. The velocity relaxation process upstream facilitates the non-uniformity and non-parallelism of the global disturbances and leads to stronger radial–axial coupling in the disturbance field at a higher ${\textit{Re}}$, showing a global dynamics beyond the ability of local analysis. Formulas of ${\textit{We}}_c$ for both Poiseuille- and uniform-velocity jets are proposed based on the results of global stability analysis. These findings elucidate the dynamics of the global instability for dripping–jetting transition under the influence of velocity relaxation and provide guidance for the precise control of jet behaviours in practical applications.

The generation and growth of wind waves are re-examined using linear viscous shear flow instability theory by solving the coupled in-air and in-water Orr–Sommerfeld equations. To enable comparison with the available laboratory observations, model simulations are performed for a wide range of wavelengths spanning the gravity–capillary and gravity wave regimes typical of such experiments. The sensitivity of the results to key modelling assumptions is investigated, including the friction velocity, the surface drift velocity at the air–water interface as well as the shapes of velocity profiles in air and in water, which are modelled using the mixing-length approach. Airflows both over an initially smooth surface and over a surface modified by the emergence of fast-growing short ripples, and thus effectively rough, are considered. A detailed energy budget analysis, based on eigenfunctions of the coupled Orr–Sommerfeld equations across different wavelengths, provides further insight into the mechanisms governing energy transfer from wind to water waves under diverse flow conditions.

Concentrated wave beams are analysed both theoretically and numerically in a general rotating and stratified axisymmetric medium, where both the rotation rate and the Brunt–Väisälä frequency vary with position. The fluid is assumed to be incompressible, weakly diffusive and weakly viscous. The analysis is conducted within the Boussinesq approximation and a linear framework, with a prescribed frequency. An asymptotic solution is derived in the limit of weak viscosity and diffusivity, describing a harmonic beam of inertia gravity waves localised around a characteristic (or ray path), similar to those generated by boundary singularities or critical points. This solution is shown to break down when the characteristic reaches a turning point which corresponds to the transition from oscillatory to evanescent behaviour. A local asymptotic analysis near the turning point demonstrates that the wave beam reflects, preserving its transverse structure while acquiring a phase shift of $\pm \pi /2$. These theoretical predictions are validated through numerical simulations, which show that the wave beam structure, both near and far from the turning point, is accurately reproduced.

To investigate how polymers influence energy transfer in three-dimensional turbulence, we conduct experiments in homogeneous bulk turbulence generated by a von Kármán swirling flow, using tomographic particle image velocimetry. A filtering approach is applied to the measured three-dimensional velocity fields to extract subgrid-scale (SGS) statistics, focusing on the filtered strain-rate tensor and SGS stress tensor. We find that polymer additives induce significant changes in the tensorial geometry: the strain-rate tensor shows a tendency towards an eigenvalue ratio of $1 : 0 : -1$, while the SGS stress tensor favours a $2 : -1 : -1$ configuration. The local energy flux – quantified by the inner product of the strain-rate and SGS stress tensors – is systematically suppressed by polymers and becomes increasingly intermittent. This suppression is linked to a reduced energy transfer efficiency, associated with the misalignment between the principal eigendirections of the two tensors. Anisotropic effects are also observed in the energy flux components, indicating that polymers affect vertical and horizontal energy transfer differently. Finally, the obtained SGS statistics allow for an a priori assessment of SGS models. Our results reveal that the nonlinear gradient model significantly outperforms the Smagorinsky model, particularly in polymer-laden turbulence. The diminished alignment between the strain-rate and SGS stress tensors may underlie the limitations of the Smagorinsky model, which assumes a scalar eddy-viscosity closure. These results provide new experimental insights into the SGS dynamics of polymeric turbulence and highlight the potential of nonlinear models for large-eddy simulations of viscoelastic flows.

Vortical flows over spinning cones with half-angles of $\theta _c =$ $10^\circ$, $15^\circ$, $22.5^\circ$, $30^\circ$ and $45^\circ$ at incidence angles of $\alpha$ between 0$^\circ$ and 36$^\circ$ are experimentally studied employing smoke streak flow visualisation and planar particle image velocimetry at Reynolds numbers of $\mathcal{O}(10^4)$ and base rotational speed ratios between 0 and 3. Symmetric vortex triads are observed on the leeward side of the stationary cones at incidence that grow in cross-section and strength along the surface. Spin breaks down this symmetry. Asymmetries in the vortex systems over the spinning cones are characterised by anti-cyclonic vortices forming in the counter-rotating meridian and cyclonic vortices in the co-rotating meridian. The anti-cyclonic vortices increase in strength as they are pushed in the direction of rotation and embrace the surface of the cone, whereas cyclonic vortices detach from the surface and exhibit unchanging vortex strength. For the most slender cone of $\theta _c = 10^\circ$, the cyclonic vortices are pushed past the plane of symmetry into the counter-rotating meridian and are squeezed between the anti-cyclonic vortex and the surface of the cone. This appears to trigger the detachment of the anti-cyclonic vortices. The thickest cone of $\theta _c = 45^\circ$ exhibits characteristics similar to flows over a disc (Kuraan & Savaş J. Visual. vol. 23, 2024, pp. 191–205), a limiting case of the cone family. As $\alpha$ increases, the stagnation point departs from the vertex and monotonically shifts along the windward surface. Regular vortex shedding events in the wake region behind the $\theta _c=45^\circ$ cone are detected in the streakline images, also a common characteristic of flows over discs. Wave patterns are observed near the leeward surface of spinning cones, which are likely signatures of the well-known centrifugal spiral wave instabilities. The bead-like features leave small-scale wave patterns on detached portions of the neighbouring trailing vortices. Inclined wave patterns form on streaklines over the entire surface of the cones, and are present in both non-spinning and spinning cases; hence, they are likely signatures of classical cross-flow boundary layer instabilities.

In this study, we explore the effect of basis functions on the performance and convergence of the Galerkin projection-based reduced-order model (ROM) in the minimal flow unit of Couette flow. POD (proper orthogonal decomposition) modes obtained from direct numerical simulation, and controllability and balanced truncation modes from the linearised Navier–Stokes equations (LNSEs) with different base flows (laminar base flow and turbulent mean flow) and an eddy viscosity model are considered. In the neighbourhood of the laminar base state, the ROMs based on the modes from the LNSEs with the laminar base flow and molecular viscosity are found to perform very well as they are able to capture the linear stability of the laminar base flow for each plane Fourier component only with a single degree of freedom. In particular, the ROM based on the balanced truncation modes models the linear dynamics involving transient growth around the laminar base flow most effectively, consistent with previous studies. In contrast, for turbulent state, the ROM based on POD modes is found to reproduce its statistics and coherent dynamics most effectively. The ROMs based on the modes from the LNSE with turbulent mean flow and an eddy viscosity model performs better compared with any other ROMs using the modes from the LNSE. These observations suggest that the performance and convergence of a ROM are highly state-dependent. In particular, this state dependence is strongly correlated with the information and dynamics that each of the basis functions contain. Discussions supporting these observations are also provided in relation to the flow physics involved and the form of coherent structures in Couette flow.

Microswimmers in suspension exhibit collective swimming behaviour, forming various self-organised structures including ordered, aggregated and turbulent-like structures. When mixed with passive particles phase-separation is known to occur, but due to the difficulty of accurately handling many-body hydrodynamic interactions, the formation of self-organised structures in mixed suspensions has remained unexplored so far. In this study, we investigate the dynamics of mixed dense suspensions of spherical bottom-heavy squirmers and obstacle spheres using Stokesian dynamics in three dimensions, taking hydrodynamic interactions into account. The results show that without an external orienting mechanism the formation of orientational order is in general disturbed by the presence of passive spheres. An initially phase-separated state is metastable for neutral or puller squirmers at high packing densities. When the squirmers are bottom-heavy, phase-separation can occur dynamically in some cases, notably as a fibrillar kind of separation for neutral squirmers and pullers at medium densities. We also observed a novel form of lamellar phase-separation for pullers at high densities with strong bottom-heaviness, with a sandwich-like structure consisting of a layer of passive particles pushed by a layer of swimmers, followed by a gap. These results indicate that microstructure and particle transport undergo significant changes depending on the type of swimmer, highlighting the importance of hydrodynamic interactions. These insights allow for a deeper understanding of the behaviour of active particles in complex fluids and to control them using external torques.

This experimental study aims to clarify how and when a weak centrifugal force affects the turbulent rotating thermal convection system. For the bulk flow, the weak centrifugal effect is significant on long-time averaged flow fields, contrasting sharply with its negligible effect on the instantaneous fields. As for the sidewall flow, it is found that properties of the boundary zonal flow are influenced by the weak centrifugal force appreciably. The onset Froude number $\textit{Fr}_c^{\textit{local}}$, signalling when the weak centrifugal effects start to set in, is found to scale with Rayleigh number $Ra$ as $\textit{Fr}_c^{\textit{local}}\sim Ra^{0.55}$ over approximately two and a half decades of $Ra$. The underlying mechanism for this robust scaling is captured by the mechanism of local force balance, which involves three unknown local scales. With the help of both a viscous–Archimedean–Coriolis argument and the experimental data, these local scales are successfully resolved to reveal a consistent result with this 0.55 scaling.

This study experimentally and numerically investigates the dynamics of a high-speed liquid jet generated from the interaction of two tandem cavitation bubbles, termed bubble 1 and bubble 2, depending on their generation sequence. Although the overall collapse pattern and jet orientation are well documented, the underlying mechanisms for supersonic jet acceleration, tip fragmentation and subsequent penetration remain to be elucidated. In our experiments, two near-identical, highly energised cavitation bubbles were generated using an underwater electric discharge method, and their transient interactions were captured using a high-speed camera. We identify three distinct jet regimes that emerge from the tip of bubble 2: conical, umbrella-shaped and spraying jets, characterised by variations in the initial bubble–bubble distance (denoted as $\gamma$) and the initiation time difference (denoted as $\theta$). Our numerical simulations using both volume of fluid and boundary integral methods reproduce the experimental observations quite well and explain the mechanism of jet acceleration. We show that the transition between the regimes is governed by the spatio-temporal characteristics of the pressure wave induced by the collapse of bubble 1, which impacts the high-curvature tip of bubble 2. Specifically, a conical jet forms when the pressure wave impacts the bubble tip prior to its contraction, while an umbrella-shaped jet develops when this impact occurs after the contraction. The spraying jets result from the breakup of the bubble tip, exhibiting mist-like and needle-like morphologies with velocities ranging from 10 to over 1200 m s−1. Remarkably, we observe that the penetration distance of spraying jets exceeds ten times the maximum bubble radius, making them ideal for long-range, controlled fluid delivery. Finally, phase diagrams for jet velocity and penetration distance in the $\gamma -\theta$ parameter space are established to provide a practical reference for biomedical applications, such as needle-free injection and micro-pumping.