A unified lattice Boltzmann method is employed to investigate Rayleigh–Bénard convection (RBC) subjected to sidewall heating and unipolar charge injection from the bottom wall. The study focuses on how the complex and nonlinear coupling between the buoyancy and Coulomb effects modify the heat transfer, flow structure and the transition between buoyancy- and Coulomb-dominated regimes. The results show that a side-heated wall, in the absence of charge injection, enhances the heat transfer rate and changes the scaling law between the Nusselt number $\textit{Nu}$ and Rayleigh number $Ra$ from $\textit{Nu} = 0.22Ra^{0.29}$ (classical RBC) to $\textit{Nu} = 0.56 Ra^{0.22}$ due to an additional buoyancy effect from the sidewall. When the electric charge is injected from the bottom wall, it is shown that the thermal boundary layer thickness decreases, leading to a further enhancement of heat transfer. Furthermore, systematic simulations over a broad range of $Ra$ and electric Rayleigh numbers $T$ reveal that, at given $T$, $\textit{Nu}$ remains constant when $Ra$ is low, indicating a Coulomb-dominated regime. Beyond a critical value of $Ra$, a power-law relationship between $\textit{Nu}$ and $Ra$ emerges, signifying a transition to the buoyancy-dominated regime. This transition can be well predicted by a dimensionless parameter, which is developed considering buoyancy to Coulomb forces. In addition, by analysing the flow structure using the Fourier mode decomposition, a phase diagram describing the dominant flow modes is proposed. The results demonstrate that the proposed dimensionless parameter not only delineates the transition between the two heat transfer regimes but also accurately captures the flow mode shift. Our findings offer new insights into the complex interaction between buoyancy and Coulomb effects and their influence on heat transfer and flow structure, with potential implications for the design of heat exchangers aimed at actively and efficiently controlling heat transfer.

We revisit the interaction of an initially uniform near-inertial wave (NIW) field with a steady background flow, with the goal of predicting the subsequent organisation of the wave field. To wit, we introduce an exact analogy between the Young–Ben Jelloul (YBJ) equation and the quantum dynamics of a charged particle in a steady electromagnetic field, whose potentials are expressed in terms of the background flow. We derive the time-averaged spatial distributions of wave kinetic energy, potential energy and Stokes drift in two asymptotic limits. In the ‘strongly quantum’ limit, where the background flow is weak compared with wave dispersion, we compute the wave statistics by extending a strong-dispersion expansion initially introduced by YBJ. In the ‘quasi-classical’ limit, where the background flow is strong compared with wave dispersion, we compute the wave statistics by leveraging the equilibrium statistical mechanics of classical systems. We compare our predictions with numerical simulations of the YBJ equation, using an instantaneous snapshot from a two-dimensional turbulent flow as the steady background flow. The agreement is very good in both limits. In particular, we quantitatively describe the preferential concentration of NIW energy in anticyclones. We predict weak NIW concentration in both asymptotic limits of weak and strong background flow, and maximal anticyclonic concentration for background flows of intermediate strength, providing theoretical underpinning to observations reported by Danioux, Vanneste and Bühler (2015 J. Fluid Mech., vol. 773, R1).

This work presents an efficient statistical model to simulate expected scalar transport in fractured porous media below the representative elementary volume scale. We focus on embedded, highly conductive, isolated fractures. The statistical integro-differential fracture model (Sid-FM) solves for ensemble-averaged solutions directly, avoiding computationally expensive Monte Carlo simulation. The expected fluid exchange between isolated fractures and the porous matrix is modelled via a non-local kernel function, leading to a set of integro-differential equations. The model is validated against reference data from Monte Carlo simulations for statistically one-dimensional test cases and shows good agreement.

Wave–sea-ice interactions shape the transition zone between open ocean and pack ice in the polar regions. Most theoretical paradigms, implemented in coupled wave–sea-ice models, predict exponential decay of the wave energy but some recent observations deviate from this behaviour. Expanding on a framework based on wave energy dissipation due to ice–water drag, we account for drifting sea ice to derive an improved model for wave energy attenuation. Analytical solutions replicate the observed non-exponential wave energy decay and the spatial evolution of the effective attenuation rate in Antarctic sea ice.

Thin liquid films play an instrumental role in the coating industry. In many cases, these films consist of multiple components and are applied in multiple layers. However, multilayer multicomponent coatings can readily develop thickness non-uniformities due to Marangoni flows driven by solute concentration gradients. Previous flow visualisation experiments have demonstrated that the addition of surfactant can suppress such non-uniformities, but the physical mechanisms underlying this suppression have not yet been definitively established. We investigate the growth of film-height non-uniformities in a two-layer multicomponent coating consisting of a solute-rich bottom layer, a solute-depleted top layer and surfactant. A lubrication-theory-based model that accounts for vertical and lateral gradients in solute and surfactant concentrations is developed. The resulting coupled nonlinear partial differential equations describing the film height, solute concentration and surfactant concentration are solved with a pseudospectral method. Our findings reveal that surfactant-induced Marangoni flows can significantly decrease film-height non-uniformities by competing with Marangoni flows due to solute concentration gradients. Several simplifications of the governing equations are explored to determine how well predictions from these simplified models compare with the full lubrication-theory-based model, thereby providing insight into dominant physical mechanisms in different parameter regimes. The role of surfactant solubility and sorption kinetics in controlling perturbation growth is also examined.

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.

A computational fluid dynamics simulation of subcooled flow boiling of water at 10.5 ${\rm bar}$, with an applied heat flux of $1\,{\rm MW}\,{\rm m}^{-2}$ and subcooling of 10 ${\rm K}$, was performed using an interface tracking method. The simulation replicated the conditions of an experiment conducted at MIT. The objectives are to elucidate heat-transfer mechanisms in moderate-pressure subcooled boiling and to validate the simulation method, with a focus on quantities that are difficult to measure experimentally, such as the distributions of velocity, temperature, bubble number density and heat-flux partitioning. Due to the small bubble size under high pressure, fine grids are required. Simulated bubble shapes, wall temperatures and vapour area fractions show good agreement with the experimental results. The simulations reveal that a very thin liquid layer (${\lt}4\,\unicode{x03BC}{\rm m}$) surrounding the bubbles is highly effective at removing heat from the surface. The local wall heat fluxes beneath medium and large bubbles, excluding the heat flux associated with seed-bubble generation, are approximately 0.9 and 0.4 ${\rm MW}\,{\rm m}^{-2}$, respectively; the latter is smaller because of the presence of thicker liquid films (14–70 $\unicode{x03BC}{\rm m}$) that thermally insulate the wall. In the single-phase liquid region, the heat transfer coefficient reaches $42\,{\rm kW}\,{\rm m}^{-2}\,{\rm K}^{-1}$ as a result of strong turbulent heat flux in the wall-normal direction; this turbulent heat flux is approximately eight times larger than in the equivalent single-phase liquid flow.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.