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.

The vertical, tip-to-tip arrangement of neighbouring caudal fins, common in densely packed fish schools, has received much less attention than staggered or side-by-side pairings. We explore this configuration using a canonical system of two trapezoidal panels (aspect ratio ${\textit{AR}}=1.2$) that pitch about their leading edges while heaving harmonically at a Strouhal number $St=0.45$ and a reduced frequency $k=2.09$. Direct numerical simulations based on an immersed-boundary method are conducted over a Reynolds-number range of $600\leq {\textit{Re}}\leq 1\times 10^{4}$, and complementary water-channel experiments extend this range to $1\times 10^{4} \leq {\textit{Re}}\leq 3\times 10^{4}$. Results indicate that when the panels oscillate in phase at a non-dimensional vertical spacing $H/c\leq 1.0$ with $c$ denoting the panel chord length, the cycle-averaged thrust of each panel rises by up to 14.5 % relative to an isolated panel; the enhancement decreases monotonically as the spacing increases. Anti-phase motion instead lowers the power consumption by up to 6 %, with only a modest thrust penalty, providing an alternative interaction regime. Flow visualisation shows that in-phase kinematics accelerate the stream between the panels, intensifying the adjacent leading-edge vortices. Downstream, the initially separate vortex rings merge into a single, larger ring that is strongly compressed in the spanwise direction; this wake compression correlates with the measured thrust gain. The interaction mechanism and its quantitative benefits persist throughout the entire numerical and experimental Reynolds-number sweep, indicating weak ${\textit{Re}}$-sensitivity within $600\leq {\textit{Re}}\leq 3\times 10^{4}$, and across multi-panel systems. These results provide the first three-dimensional characterisation of tip-to-tip flapping-panel interactions, establish scaling trends with spacing and phase, and offer a reference data set for reduced-order models of vertically stacked propulsors.

This paper considers the problem of water wave scattering by a rectangular anisotropic elastic plate mounted on the ocean surface, with either free, clamped or simply supported edges. The problem is obtained as an expansion over the dry modes of the elastic plate, which are computed using a Rayleigh–Ritz method. In turn, the component diffraction and radiation problems are solved by formulating a boundary integral equation and solving numerically using a constant panel method. The results are presented to highlight the resonant responses of the plate under different forcing scenarios. In particular, we illustrate how the excitation of certain modes can be forbidden due to symmetry.

Many species of fish, as well as biorobotic underwater vehicles (BUVs), employ body–caudal fin (BCF) propulsion, in which a wave-like body motion culminates in high-amplitude caudal fin oscillations to generate thrust. This study uses high-fidelity simulations of a mackerel-inspired caudal fin swimmer across a wide range of Reynolds and Strouhal numbers to analyse the relationship between swimming kinematics and hydrodynamic forces. Central to this work is the derivation and use of a model for the leading-edge vortex (LEV) on the caudal fin. This vortex dominates the thrust production from the fin and the LEV model forms the basis for the derivation of scaling laws grounded in flow physics. Scaling laws are derived for thrust, power, efficiency, cost-of-transport and swimming speed, and are parametrised using data from high-fidelity simulations. These laws are validated against published simulation and experimental data, revealing several new kinematic and morphometric parameters that critically influence hydrodynamic performance. The results provide a mechanistic framework for understanding thrust generation, optimising swimming performance, and assessing the effects of scale and morphology in aquatic locomotion of both fish and BUVs.

Dense granular flows exhibit pronounced non-local behaviours, particularly in creeping regions and shear-localised zones, which challenges classical local inertial rheologies. In this work, we develop a continuum framework for dense granular flows by extending the $\mu (I)$ rheology through the inclusion of granular temperature as an explicit state variable, thereby establishing a direct link between grain-scale velocity fluctuations and macroscopic stresses, and enabling the representation of non-local effects. The model is implemented within a finite-volume computational framework, and systematically validated against three canonical configurations spanning steady and transient regimes: heap flows, split-bottom Couette flows, and granular column collapse. Across these benchmarks, the formulation captures key non-local features observed experimentally and numerically, including sustained creeping below yield, shear-band broadening and migration, and the transient evolution of free surfaces and runout dynamics. Overall, the granular-temperature-extended $\mu (I)$ rheology provides a unified continuum description that reconciles local and non-local behaviour in dense granular flows, retains the predictive capability of inertial rheology in rapid regimes, and extends its applicability to creeping and shear-localised flows. The proposed framework offers a physically interpretable and scalable basis for modelling granular processes in both geophysical and industrial contexts.

A classical and central problem in the theory of water waves is to classify parameter regimes for which non-trivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number $ \textit{Fr}$ (a non-dimensional wave speed) plays the central role. So far, the best analytical result $ \textit{Fr} \lt \sqrt {2}$ was obtained by Starr (1947 J. Mar. Res., vol. 6, pp. 175–193), while the numerical evidence of Longuet-Higgins & Fenton (1974 Proc. A, vol. 340, pp. 471–493) states $ \textit{Fr} \leq 1.294$. On the other hand, as shown recently by Kozlov (2023 On the first bifurcation of Stokes waves), the hypothetical upper bound $ \textit{Fr} \lt 1.399$ is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy and rigorously establish the improved upper bound $ \textit{Fr} \lt 1.3451$, which is the first rigorous improvement of Starr’s bound. In this process, we establish several new inequalities for the relative horizontal velocity, which are of separate interest and for which we delicately make use of the bound on the slope of the surface profile established by Amick (1987 Arch. Ration. Mech. Anal., vol. 99, pp. 91–114). As an application we show that the velocity at the bottom below the crest of any solitary wave does not exceed $47\,\%$ of the propagation speed.

While studying soap film bursting to validate their opening velocity, i.e. the Taylor–Culick velocity, Mysels and co-workers discovered fifty years ago a compression region propagating in front of the hole that they called the aureole. In the wake of such a discovery, a series of papers ‘Bursting of soap films’ focused on the study of such peculiar Marangoni flow resulting from the rapid surfactant compression. Their pioneering theory postulates that surfactants remain insoluble at the interface, leading to a self-similar process that has been verified on small films. In the present study, by using films large enough to allow the surfactant to relax, we reveal a previously unexplored regime of aureole development. The surfactants forming the aureole initially behave as if they were insoluble, with an aureole front propagating at a constant speed. After a few milliseconds, however, the front slows down until it matches the hole-opening velocity, and the aureole length then becomes constant. In this steady regime, a model taking into account surfactant advection/diffusion in the film is developed. Our theory accurately captures the thickness and velocity exponential profiles observed in experiments, demonstrating that the observed deviations arise from a balance between the surfactant rapid compression and a desorption flux. Furthermore, measurements of the characteristic aureole lengths provide estimates of physico-chemical properties of the monolayer, which are discussed in the light of predictions based on adsorption laws. The present study highlights the transition from the insoluble limit to the soluble limit, and paves the way for measurement of out-of-equilibrium dynamics of surfactants.

We develop a weakly nonlinear model of duct acoustics in two and three dimensions (without flow). The work extends the previous work of McTavish & Brambley (2019 J. Fluid Mech., vol. 875, pp. 411–447) to three dimensions and significantly improves the numerical efficiency. The model allows for general curvature and width variation in two-dimensional ducts, and general curvature and torsion with radial width variation in three-dimensional ducts. The equations of gas dynamics are perturbed and expanded to second order, allowing for wave steepening and the formation of weak shocks. The resulting equations are then expanded temporally in a Fourier series and spatially in terms of straight-duct modes, and a multi-modal method is applied, resulting in an infinite set of coupled ordinary differential equations for the modal coefficients. A linear matrix admittance and its weakly nonlinear generalisation to a tensor convolution are first solved throughout the duct, and then used to solve for the acoustic pressures and velocities. The admittance is useful in its own right, as it encodes the acoustic and weakly nonlinear properties of the duct independently from the specific wave source used. After validation, a number of numerical examples are presented that compare two- and three-dimensional results, the effects of torsion, curvature and width variation, acoustic leakage due to curvature and nonlinearity and the variation in effective duct length of a curved duct due to varying the acoustic amplitude. The model has potential future applications to sound in brass instruments. Matlab source code is provided in the supplementary material.

Free-surface cusps are a generic feature of externally driven, viscous flow bounded by a free surface, in that their form is stable under small perturbations. Here we present an alternative to the boundary integral description found recently (J. Eggers, Phys. Rev. Fluids, vol. 8, 2023, 124001), which is based directly on a local analysis of the Stokes equation. The new description has the advantage of greater simplicity and transparency, allowing us to understand the connections with bifurcation theory, as well as with other physical systems displaying similar singularities. To illustrate this, we construct cusp solutions corresponding to higher-order singularities, as well as time-dependent solutions.

We develop, simulate and extend an initial proposition by Chaves et al. (J. Stat. Phys., vol. 113, no. 5-6, 2003, pp. 643–692) concerning a random incompressible vector field able to reproduce key ingredients of three-dimensional turbulence in both space and time. In this paper we focus on the important underlying Gaussian framework. Presently, the statistical spatial structure of this velocity field is consistent with a divergence-free fractional Gaussian vector field that encodes all known properties of homogeneous and isotropic fluid turbulence at a given finite Reynolds number, up to second-order statistics. The temporal structure of the velocity field is introduced through a stochastic evolution of the respective Fourier modes. In the simplest picture, Fourier modes evolve according to an Ornstein–Uhlenbeck process, where the characteristic time scale depends on the wave-vector amplitude. For consistency with direct numerical simulations (DNS) of the Navier–Stokes equations, this time scale is inversely proportional to the wave-vector amplitude. As a consequence, the characteristic velocity that governs the eddies is independent of their size and is related to the velocity standard deviation, which is consistent with some features of the so-called sweeping effect. To ensure differentiability in time while respecting the Markovian nature of the evolution, we use the methodology developed by Viggiano et al. (J. Fluid Mech., vol. 900, 2020, A27) to propose a fully consistent stochastic picture. We finally derive analytically all statistical quantities in a continuous set-up and develop precise and efficient numerical schemes of the corresponding periodic framework. Both exact predictions and numerical estimations of the model are compared with DNS provided by the Johns Hopkins database.

A data-driven algorithm is proposed that employs sparse data from velocity and/or scalar sensors to forecast the future evolution of three-dimensional turbulent flows. The algorithm combines time-delayed embedding together with Koopman theory and linear optimal estimation theory. It consists of three steps: dimensionality reduction, currently with proper orthogonal decomposition (POD); construction of a linear dynamical system for current and future POD coefficients; and system closure using sparse sensor measurements. In essence, the algorithm establishes a mapping from current sparse data to the future state of the dominant structures of the flow over a specified time window. The method is scalable (i.e. applicable to very large systems), physically interpretable and provides sequential forecasting on a sliding time window of prespecified length. It is applied to the turbulent recirculating flow over a surface-mounted cube (with more than $10^8$ degrees of freedom) and is able to forecast accurately the future evolution of the most dominant structures over a time window at least two orders of magnitude larger that the (estimated) Lyapunov time scale of the flow. Most importantly, increasing the size of the forecasting window only slightly reduces the accuracy of the estimated future states.

This study utilises chromocapillary stresses induced by light-actuated photosurfactants to demonstrate theoretically that a stable uniform liquid layer wetting a substrate can be sculpted and stirred on the microscale. A mathematical model is presented for two photosurfactant species that can switch from trans to cis states. Switching takes place in the bulk and on the interface, and convection–diffusion–reaction equations describe the local concentrations there. Under uniform light illumination (e.g. blue light) the equilibrium concentrations of trans and cis are non-uniform with layer depth, and a quiescent state with a flat interface exists. A non-uniform light intensity along the layer is superimposed to drive the system out of equilibrium, and induce interfacial deformations and flow in the bulk. This is carried out asymptotically for small-intensity non-uniformities and the first-order non-uniform solutions are found in semi-analytic form. The solutions show that a local increase in intensity increases the surface tension locally by sweeping surfactant off the interface to generate an inward trapping flow (known as a ‘Marangoni tweezer’ in experiments). Light intensities with a sinusoidal variation along the interface are also considered to show that vortical mixing motions are set up. Additionally, the liquid sculpting problem is analysed and a class of inverse problems are solved to predict the distribution of the light intensity required to produce a desired target interfacial shape. Finally, a parametric study is carried out to evaluate the effect of Biot, Damköhler and Marangoni numbers on the maximum light-induced interfacial velocity.