We simulate thermal convection in a two-dimensional square box using the no-slip condition on all boundaries, and isothermal bottom and top walls, and adiabatic sidewalls. We choose 0.1 and 1 for the Prandtl number $Pr$ and vary the Rayleigh number $Ra$ between $10^6$ and $10^{12}$. We particularly study the temporal evolution of integral transport quantities towards their steady states. Perhaps not surprisingly, the velocity field evolves more slowly than the thermal field, and its steady state – which is nominal in the sense that large-amplitude low-frequency oscillations persist around plausible averages – is reached exponentially. We study these oscillation characteristics. The transient time for the velocity field to achieve its nominal steady state increases almost linearly with the Reynolds number. For large $Ra$, the Reynolds number itself scales almost as $Ra^{2/3}\, Pr^{-1}$, and the Nusselt number as $Ra^{2/7}$.

An analytical formulation is provided that describes the first two natural modes of the fluid–structure interaction of an incompressible current with a pitching and heaving flexible plate. The objective is twofold: first, to present a general derivation of analytical expressions for the lift, moment and the flexural moments exerted by an inviscid flow on a pitching and heaving plate whose deformation is general enough that the coupling of the flexural moments with the structural equations allows solving analytically the first two natural modes of the system; second, to analyse the propulsion performance of the foil when actuated near the first two natural frequencies. For the second purpose, one also needs the thrust force generated through the motion and the general deformation of the foil considered, which is analytically derived using the linearized vortex impulse theory, extending and systematizing previous works. The analytical expressions, once viscous effects are taken into consideration through nonlinear transverse damping and offset drag coefficients, are compared with small-amplitude available experimental data, discussing their limitations. It is found that low stiffness pitching and heaving are quite different, with a pitching flexible foil only generating thrust near the second resonant frequency, whereas heaving always generates thrust, with the maximum slightly below the second natural frequency. Maximum thrust for large stiffness pitching is around the first natural frequency. The maximum efficiency occurs at frequencies close to the first natural mode if the foil is sufficiently rigid, but it is not related to the natural frequencies as the rigidity decreases.

Rayleigh–Taylor (RT) stability occurs when a single-mode light/heavy interface is accelerated by rarefaction waves, exhibiting a sustained oscillation in perturbation amplitude. If the perturbation is accelerated again by a shock propagating in the same direction as the rarefaction waves, the interface evolution will shift from RT stability to Richtmyer–Meshkov (RM) instability. Depending upon the interface state when the shock arrives, the perturbation growth can be actively manipulated through controlling the magnitudes of vorticity deposited by rarefaction and shock waves. The present work first theoretically analyses the 12 different growth possibilities of a light/heavy interface accelerated by co-directional rarefaction and shock waves. A theoretical model is established by combining the RT growth rate with the RM growth rate, providing the conditions for the different possibilities of the perturbation growth. Based on the model, extensive experiments are designed and conducted in the specially designed rarefaction-shock tube. By precisely controlling the shock arrival time at the interface, the different growth possibilities, including promotion, reduction and freeze-out, are realised in experiments. This work verifies the feasibility of manipulating the light/heavy perturbation via co-directional rarefaction and shock waves, which sheds light on control of hydrodynamic instabilities in practical applications.

This paper numerically investigates the heat transport and bifurcation of natural convection in a differentially heated cavity filled with entangled polymer solution combined with the boundary layer and kinetic energy budget analysis. The polymers are described by the Rolie-Poly model, which effectively captures the rheological response of entangled polymers. The results indicate that the competition between its shear-thinning and elasticity dominates the flow structures and heat transfer rate. The addition of polymers tends to enhance the heat transfer as the polymer viscosity ratio ($\beta$) decreases or the relaxation time ratio ($\xi$) increases. The amount of heat transfer enhancement (HTE) behaves non-monotonically, which first increases significantly and then remains almost constant or decreases slightly with the Weissenberg number ($Wi$). The critical $Wi$ gradually increases with the increasing $\xi$, where the maximum HTE reaches approximately $64.9\,\%$ at $\beta = 0.1$. It is interesting that even at low Rayleigh numbers, the flow transitions from laminar to periodic flows in scenarios with strong elasticity. The bifurcation is subcritical and exhibits a typical hysteresis loop. Then, the bifurcation routes driven by inertia and elasticity are examined by direct numerical simulations. These results are illustrated by time histories, Fourier spectra analysis and spatial structures observed at varying time intervals. The kinetic energy budget indicates that the stretch of the polymers leads to great energy exchange between polymers and flow structures, which plays a crucial role in the hysteresis phenomenon. This dynamic behaviour contributes to the strongly self-sustained and self-enhancing processes in the flow.

Periodic travelling waves at the free surface of an incompressible inviscid fluid in two dimensions under gravity are numerically computed for an arbitrary vorticity distribution. The fluid domain over one period is conformally mapped from a fixed rectangular one, where the governing equations along with the conformal mapping are solved using a finite-difference scheme. This approach accommodates internal stagnation points, critical layers and overhanging profiles, thereby overcoming limitations of previous studies. The numerical method is validated through comparisons with known solutions for zero and constant vorticity. Novel solutions are presented for affine vorticity functions and a two-layer constant-vorticity scenario.

Generation of steady streaming vortices is usually accomplished by mechanically vibrating bodies, as occurs in several microfluidic applications for mixing, as well as for transport and handling of microparticles. Here, we propose the generation of streaming from the harmonic electromagnetic forcing of a free-moving circular magnet held afloat on a shallow electrolytic layer, and show that the sense of rotation of steady vortices is the opposite to that of the classical streaming flow. Reverse streaming is attributed to the coupling between the fluid and the free-moving body. Analytical solutions offer a physical rationale for the observed flow dynamics, while numerical simulation reproduces the experimental observations satisfactorily.

Phase change materials (PCMs) hold considerable promise for thermal energy storage applications. However, designing a PCM system to meet a specific performance presents a formidable challenge, given the intricate influence of multiple factors on the performance. To address this challenge, we hereby develop a theoretical framework that elucidates the melting process of PCMs. By integrating stability analysis with theoretical modelling, we derive a transition criterion to demarcate different melting regimes, and subsequently formulate the melting curve that uniquely characterises the performance of an exemplary PCM system. This theoretical melting curve captures the key trends observed in experimental and numerical data across a broad parameter space, establishing a convenient and quantitative relationship between design parameters and system performance. Furthermore, we demonstrate the versatility of the theoretical framework across diverse configurations. Overall, our findings deepen the understanding of thermo-hydrodynamics in melting PCMs, thereby facilitating the evaluation, design and enhancement of PCM systems.

In this work, we study the effect of flow curvature, or angular momentum, on the propagation and trapping characteristics of near-inertial waves (NIWs) in a curved front. The curved front is idealised as a baroclinic vortex in cyclogeostrophic balance. Motivated by ocean observations, we employ a Gaussian base flow, which by construction possesses a shield of oppositely signed vorticity surrounding its core, and we consider both cyclonic and anticyclonic representations of this flow. Following two main assumptions, i.e. that (i) the horizontal wavelength of the NIW is smaller than the length scale of the background flow (the WKBJ approximation), and (ii) the vertical wavelength of the NIW is smaller than the radial distance of interest, we derive the NIW dispersion relation and discuss the group velocity and direction of energy propagation. We show that the curvature can (i) increase the critical depth and horizontal extent of the trapping region, (ii) reduce NIW activity at the centre of the anticyclonic vortex core and enhance it in the cyclonic shield surrounding the core for high curvatures, (iii) lead to NIW trapping in the anticyclonic shield surrounding the cyclonic core, and (iv) increase the available band of NIW frequencies that are trapped. The solutions from the ray-tracing method are supported by numerical solutions of the governing equations linearised about the cyclogeostrophic base state. Finally, these methods are applied to an idealised model of oceanic mesoscale Arctic eddies showing an increase in the critical depth of trapping. Our results – while applied to polar eddies – equally apply at lower latitudes in both oceans and atmospheres, highlighting the potential importance of flow curvature in controlling the propagation of NIW energy.

This paper presents a theoretical investigation of vortex modes in acoustofluidic cylindrical resonators with rigid boundaries and viscous fluids. By solving the Helmholtz equation for linear pressure, incorporating boundary conditions that account for no-slip surfaces and vortex and non-vortex excitation at the base, we analyse both single- and dual-eigenfunction modes near system resonance. The results demonstrate that single-vortex modes generate spin angular momentum exclusively along the axial direction, while dual modes introduce a transverse spin component due to the nonlinear interaction between axial and transverse ultrasonic waves, even in the absence of vortex excitation. We find that nonlinear acoustic fields, including energy density, radiation force potential and spin, scale with the square of the shear wave number, defined as the ratio of the cavity radius to the thickness of the viscous boundary layer. Theoretical predictions align closely with finite element simulations based on a model for an acoustofluidic cavity with adiabatic and rigid walls. These findings hold particular significance for acoustofluidic systems, offering potential applications in the precise control of cells and microparticles.

Direct numerical simulations have been conducted to explore the coupling effect of the thermoelectric effect and vertical convection (VC) in a square cavity composed of liquid lithium and stainless steel under different Hartmann numbers at $Ra=10^5$. By leveraging thermoelectric phenomena, an innovative approach is proposed to actively modulate heat transfer efficiency. The core concept lies in modulating the intensity of large-scale circulation (LSC) in VC systems through the torque generated by the interaction between thermoelectric currents and magnetic fields via Lorentz forces. The findings reveal that when the torque aligns with the direction of LSC induced by pure buoyancy, both momentum and heat transfer are enhanced. However, due to the magnetic damping itself, this enhancement is not sustained indefinitely, resulting in a trend of initial increase followed by decline in both momentum and heat transfer efficiency. Conversely, when the magnetic field direction is reversed, causing the Lorentz force torque to oppose the buoyancy-driven circulation, both momentum and heat transfer efficiency diminish until the flow reverses. By varying the magnetic field intensity, three distinct flow regimes are identified: the buoyancy-dominated regime, the thermoelectric-dominated regime and the magnetic-damping-dominated regime. The transition between the buoyancy-dominated regime and thermoelectric-dominated regime – specifically, the onset of flow reversal – is analysed through a boundary-layer–bulk–boundary-layer coupling model. This model enables precise prediction of the critical $Ha$ based on the torque balance between buoyancy forces and thermoelectrically induced Lorentz forces, and demonstrates close agreement with numerical simulations.

Ocean submesoscales, flows with characteristic size $10\,\text{m}{-}10\,\text{km}$, are transitional between the larger, rotationally constrained mesoscale and three-dimensional turbulence. In this paper, we present simulations of a submesoscale ocean filament. In our case, the filament is strongly sheared in both vertical and cross-filament directions, and is unstable. Instability indeed dominates the early behaviour with a fast extraction of kinetic energy from the vertically sheared thermal wind. However, the instability that emerges does not exhibit characteristics that match the perhaps expected symmetric or Kelvin–Helmholtz instabilities, and appears to be non-normal in nature. The prominence of the transient response depends on the initial noise, and for large initial noise amplitudes, saturates before symmetric instability normal modes are able to develop. The action of the instability is sufficiently rapid – with energy extraction from the mean flow emerging and peaking within the first inertial period ($\sim\! 18\ \text{h}$) – that the filament does not respond in a geostrophically balanced sense. Instead, at all initial noise levels, it later exhibits vertically sheared near-inertial oscillations with higher amplitude as the initial minimum Richardson number decreases. Horizontal gradients strengthen only briefly as the fronts restratify. These unstable filaments can be generated by strong mixing events at pre-existing stable structures; we also caution against inadvertently triggering this response in idealised studies that start in a very unstable state.

Bubbles entrained by breaking waves rise to the ocean surface, where they cluster before bursting and release droplets into the atmosphere. The ejected drops and dry aerosol particles, left behind after the liquid drop evaporates, affect the radiative balance of the atmosphere and can act as cloud condensation nuclei. The remaining uncertainties surrounding the sea spray emissions function motivate controlled laboratory experiments that directly measure and link collective bursting bubbles and the associated drops and sea salt aerosols. We perform experiments in artificial seawater for a wide range of bubble size distributions, measuring both bulk and surface bubble distributions (measured radii from $30\,\unicode{x03BC} \mathrm{m}$ to $5\,\mathrm{mm}$), together with the associated drop size distribution (salt aerosols and drops of measured radii from $50\,\mathrm{nm}$ to $500\,\unicode{x03BC} \mathrm{m}$) to quantify the link between emitted drops and bursting surface bubbles. We evaluate how well the individual bubble bursting scaling laws describe our data across all scales and demonstrate that the measured drop production by collective bubble bursting can be represented by a single framework integrating individual bursting scaling laws over the various bubble sizes present in our experiments. We show that film drop production by bubbles between $100\,\unicode{x03BC} \mathrm{m}$ and $1\,\mathrm{mm}$ describes the submicron drop production, while jet drop production by bubbles from $30\,\unicode{x03BC} \mathrm{m}$ to $2\,\mathrm{mm}$ describes the production of drops larger than $1\,\unicode{x03BC} \mathrm{m}$. Our work confirms that sea spray emission functions based on individual bursting processes are reasonably accurate as long as the surface bursting bubble size distribution is known.

We present the results of a theoretical investigation of orbital stability in pilot-wave hydrodynamics, wherein a droplet bounces and self-propels across the surface of a vertically vibrating liquid bath. A critical notion in pilot-wave hydrodynamics is that the bath plays the role of the system memory, recording the history of the droplet in its wave field. Quantised orbital motion may arise when the droplet is confined by either an axisymmetric potential or the Coriolis force induced by system rotation. We here elucidate the dependence of the stability of circular orbits on both the form of the confining force and the system memory. We first provide physical insight by distinguishing between potential- and wave-driven instabilities. We demonstrate that the former are a generic feature of classical orbital dynamics at constant speed, while the latter are peculiar to pilot-wave systems. The wave-driven instabilities are marked by radial perturbations that either grow monotonically or oscillate at an integer multiple of the orbital frequency, in which case they are said to be resonant. Conversely, for potential-driven wobbling, the instability frequency may be resonant or non-resonant according to the form of the applied potential. Asymptotic analysis rationalises the different stability characteristics for linear-spring and Coriolis forces, the two cases that have been explored experimentally. Our results are generalised to consider other potentials of interest in pilot-wave hydrodynamics, and elucidate the distinct roles of wave- and potential-driven instabilities. Our study highlights the limitations of prior heuristic arguments for predicting the onset of orbital instability.

The Leidenfrost effect occurs when drops are deposited on a highly superheated solid surface, creating a thin vapour film through rapid evaporation that levitates the drops. For drop with a radius exceeding a critical value, a vapour bubble forms and bursts from its bottom centre, a phenomenon known as chimney instability. Despite extensive investigation, the impact of Leidenfrost drop’s rotation on its chimney instability has remained unexplored. This study addresses this gap by providing both numerical and approximate solutions to the theoretical models. We identify two distinct regimes where either gravitational force or centrifugal force is the primary driver of chimney instability. These regimes are characterised by a non-dimensional rotation number, Ro, which represents the ratio between centrifugal force and gravitational force. Our findings reveal clear scaling laws that relate the critical geometrical parameters (radius, volume and height of the drop) for chimney instability to Ro, demonstrating that rotation can induce chimney instability in smaller drops. The scaling laws are elucidated through pressure perturbation analyses under a virtual perturbation to the drop profile at the critical state for chimney instability. Additionally, by varying the evaporation number $Ev$, we demonstrate that while increased superheat reduces the critical radius in the absence of rotation, the scaling laws related to Ro for a rotating drop remain unaffected. Building on these insights, we present a master curve in a simplified form that accurately predicts the critical state for chimney instability under various angular velocities, gravitational accelerations and superheat conditions.