The shallow-water equations are widely used to model interactions between horizontal shear flows and (rotating) gravity waves in thin planetary atmospheres. Their extension to allow for interactions with magnetic fields – the equations of shallow-water magnetohydrodynamics (SWMHD) – is often used to model waves and instabilities in thin stratified layers in stellar and planetary atmospheres, in the perfectly conducting limit. Here we consider how magnetic diffusion should be added to the equations of SWMHD. This is crucial for an accurate balance between advection and diffusion in the induction equation, and hence for modelling instabilities and turbulence. For the straightforward choice of Laplacian diffusion, we explain how fundamental mathematical and physical inconsistencies arise in the equations of SWMHD, and show that unphysical dynamo action can result. We then derive a physically consistent magnetic diffusion term by performing an asymptotic analysis of the three-dimensional equations of magnetohydrodynamics in the thin-layer limit, giving the resulting diffusion term explicitly in both planar and spherical coordinates. We show how this magnetic diffusion term, which allows for a horizontally varying diffusivity, is consistent with the standard shallow-water solenoidal constraint, and leads to negative semidefinite Ohmic dissipation. We also establish a basic type of antidynamo theorem.

We examine the separate effects of turbulence beneath a free surface and non–breaking surface capillary waves on the gas-transfer velocity of atmospheric oxygen into water across an air–water interface. The experiments are conducted in a recirculating open water channel with quiescent air, where atmospheric oxygen naturally dissolves into the water via the exposed surface. Through the combination of an active turbulence grid and an array of surface penetrating dowels, we are able to separate the effects of sub-surface turbulence and surface capillary waves. The findings demonstrate that the gas-transfer velocity trends with the turbulence properties, not the capillary wave properties, thus indicating that, when both are present, it is the sub-surface turbulence, not the capillary waves, that plays the dominant role in determining the rate of gas transfer across an air–water interface in the non-breaking capillary wave regime.

Laminar–turbulent transition on the suction surface of the LM45.3p blade ($20\,\%$ thickness) was investigated using wall-resolved large eddy simulation (LES) at a chord Reynolds number of $Re_c=10^6$ and angle of attack $4.6^\circ$. The effects of anisotropic free stream turbulence (FST) with intensities $TI=0\,\%$–$7\,\%$ were examined, with integral length scales scaled down from atmospheric measurements. At $TI=0\,\%$, a laminar separation bubble (LSB) forms and transition is initiated by Kelvin–Helmholtz vortices. At low FST levels ($0\,\%\lt TI \leqslant 2.4\,\%$), robust streak growth via the lift-up mechanism suppresses the LSB, while transition dynamics shifts from two-dimensional Tollmien–Schlichting (TS) waves ($TI=0.6\,\%$) to predominantly varicose inner and outer instabilities ($TI=1.2\,\%$ and $2.4\,\%$) induced by the wall-normal shear and inflectional velocity profiles. The critical disturbance kinetic energy scales with $TI^{-1.80\pm 0.11}$, compared with $TI^{-2.40}$ from Mack’s correlation. For $TI\geqslant 4.5\,\%$, bypass transition dominates, driven by high-frequency boundary layer perturbations and streak breakdown via outer sinuous modes induced by the spanwise shear and inflectional velocity profiles. The scaling of streak amplitudes with $TI$ becomes sub-linear and spanwise non-uniformity characterises the turbulent breakdown. The critical disturbance kinetic energy reduces to $TI^{-0.90\pm 0.16}$, marking a transition regime distinct from modal mechanisms. The onset of bypass transition ($TI\approx 2.4\,\%{-}4.5\,\%$) aligns with prior studies of separated and flat-plate flows. A proposed turbulence spectrum cutoff links atmospheric measurements to wind tunnel data and Mack’s correlation, offering a framework for effective $TI$ estimation in practical environments.

We address the problem of shock-induced ignition and transition to detonation in a reactive medium in the presence of mechanically induced fluctuations by a moving oscillating piston. For the inert problem prior to ignition, we provide a novel closed-form model in Lagrangian coordinates for the generation of the train of compression and expansions, their steepening into a train of N-shock waves and their reflection on the lead shock, as well as the distribution of the energy dissipation rate in the induction zone. The model is found to be in excellent agreement with numerics. Reactive calculations were performed for hydrogen and ethylene fuels using a novel high-fidelity scheme to solve the reactive Euler equations written in Lagrangian coordinates. Different regimes of ignition and transition to detonation, controlled by the time scale of the forcing and the two time scales of the chemistry: the induction and reaction times. Two novel hotspot cascade mechanisms were identified. The first relies on the coherence between the sequence of hotspot formation set by the piston forcing and forward-wave interaction with the lead shock, generalising the classic runaway in fast flames. The second hotspot cascade is triggered by the feedback between the pressure pulse generated by the first-generation hotspot cascade and the shock. For slow forcing, the sensitisation is through a modification to the classic runaway process, while the high-frequency regime leads to very localised subcritical hotspot formation controlled by the cumulative energy dissipation of the first-generation shocks at a distance comparable with the shock formation location.

The Hasselmann equation for the nonlinear interactions of deep-water gravity waves differs from other four-wave kinetic equations by the interaction coefficient. The explicit formula for this coefficient (e.g. Krasitskii, J. Fluid. Mech., vol. 272, 1994, pp. 1–20; Zakharov, Eur. J. Mech. B/Fluids, vol. 18. issue 3, 1999, pp. 327–344) is of great complexity and leaves its properties obscured. We provide analytical results for the behaviour of the coefficient in different domains. The Phillips curve and discrete interaction approximation-like quadruplets are studied in detail. The coupling coefficient for the long–short wave interactions is calculated and found to be surprisingly small. This smallness greatly reduces the non-locality of the interactions.

This study investigates the hydrodynamic interaction between a fully submerged buoyant pendulum and surface gravity waves, focusing on its primary and subharmonic resonance behaviour. The oscillatory motion of the pendulum is driven by fluid drag, with primary resonance occurring at the forcing frequency (viz. the wave frequency) and subharmonic resonance manifesting at half the forcing frequency. Both resonances exhibit nonlinear characteristics, including jump-up, jump-down phenomena and hysteresis. Furthermore, particle image velocimetry results reveal that the velocity fields of the surrounding fluid oscillate at the forcing frequency, confirming that subharmonic resonance is not induced by subharmonic excitation within the velocity field. Experimental observations are validated through both analytical and numerical methods, particularly within the primary and subharmonic resonance frequency ranges. The theoretical model describes the transverse motion of the pendulum using a nonlinear ordinary differential equation, with the method of multiple scales employed for the analytical solution. These analyses reveal the nonlinear characteristics of the system, e.g. bistable response of the primary/subharmonic resonances, and identify three distinct response regions based on the forcing frequency and amplitude. The system exhibits primary resonance regardless of the excitation strength; however, an unstable solution arises if the excitation level surpasses a specific threshold value. In contrast, subharmonic resonance is triggered only when the excitation amplitude exceeds a critical value. Furthermore, the experimental hysteresis curve confirms the theoretically predicted primary and subharmonic resonances, along with the jump-up and jump-down characteristics.

The helicity is a topological conserved quantity of the Euler equations which imposes significant constraints on the dynamics of vortex lines. In the compressible setting, the conservation law holds only under the assumption that the pressure is barotropic. Let us consider a volume $V$ containing a compressible fluid with density $\rho$, velocity field $\textbf{u}$ and vorticity $\boldsymbol{\omega}$. We show that by introducing a new definition of helicity density $h_{\rho }=(\rho {\boldsymbol {u}})\cdot \mbox {curl}\,(\rho {\boldsymbol {u}})$ the barotropic assumption on the pressure can be removed, although ${\int _{V}} h_{\rho }{\rm d}V$ is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density $h_{\rho }$ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux ${\boldsymbol {J}}_{\rho }$ contains all the pressure terms and whose source involves the potential vorticity $q = \boldsymbol{\omega} \cdot \nabla \rho$. Therefore, the rate of change of ${\int _{V}} h_{\rho }{\rm d}V$ no longer depends on the pressure and is easier to analyse, as it depends only on the potential vorticity and kinetic energy as well as $\mbox {div}\,{\boldsymbol {u}}$. This result also carries over to the inhomogeneous incompressible Euler equations for which the potential vorticity $q$ is a material constant. Therefore, $q$ is bounded by its initial value $q_{0}=q({\boldsymbol {x}},\,0)$, which enables us to define an inverse resolution length scale $\lambda _{H}^{-1}$ whose upper bound is found to be proportional to $\|q_{0}\|_{\infty }^{2/7}$. In a similar manner, we also introduce a new cross-helicity density for the ideal non-barotropic magnetohydrodynamic (MHD) equations.

Attenuation of shock waves through dense granular media with varying macro-scale and micro-scale parameters has been numerically studied in this work by a coupled Eulerian–Lagrangian approach. The results elucidate the correlation between the attenuation mechanism and the nature of shock-induced unsteady flows inside the granular media. As the shock transmission becomes trivial relative to the establishment of unsteady interpore flows, giving way to the gas filtration, the shock attenuation mechanism transitions from the shock dynamics and deduction of propagation area associated with the shock transmission, to the drag-related friction dissipation alongside the gas filtration. The ratio between the maximum shock transmission length and the thickness of the particle layer is found to be a proper indicator of the nature of shock-induced flows. More importantly, it is this ratio that successfully collapses the upstream and downstream pressures of shock impacted particle layers with widely ranging thickness and volume fraction, leading to a universal scaling law for the shock attenuation effect. We further propose a correlation between the structure of particle layer and the corresponding maximum shock transmission length, guaranteeing adequate theoretical predictions of the upstream and downstream pressures. These predictions are also necessary for an accurate estimation of the spread rate of shock dispersed particle bed through a pressure-gradient-based scaling method.

When a drop impacts a solid substrate or a thin liquid film, a thin gas disc is entrapped due to surface tension, the gas disc retracting into one or several bubbles. While the evolution of the gas disc for impact on solid substrate or film of the same fluid as the drop has been largely studied, little is known on how it varies when the liquid of the film is different from that of the drop. We study numerically the latter unexplored area, focusing on the contact between the drop and the film, leading to the formation of an air bubble. The volume of fluid method was adapted to three fluids in the framework of the Basilisk solver. The numerical simulations show that the deformation of the liquid film due to air cushioning plays a crucial role in bubble entrapment. A new model for the contact time and the entrapment geometry was deduced from the case of the impact on a solid substrate. This was done by considering the deformation of the thin immiscible liquid layer during impact depending mainly on its thickness and viscosity. The lubrication of the gas layer was found to be the major effect governing bubble entrapment. However, the film viscosity was also identified as having a critical role in bubble formation and evolution; the magnitude of its influence was also quantified.

Gravity currents are a ubiquitous density-driven flow occurring in both the natural environment and in industry. They include: seafloor turbidity currents, primary vectors of sediment, nutrient and pollutant transport; cold fronts; and hazardous gas spills. However, while the energetics are critical for their evolution and particle suspension, they are included in system-scale models only crudely, so we cannot yet predict and explain the dynamics and run-out of such real-world flows. Herein, a novel depth-averaged framework is developed to capture the evolution of volume, concentration, momentum and turbulent kinetic energy from direct integrals of the full governing equations. For the first time, we show the connection between the vertical profiles, the evolution of the depth-averaged flow and the energetics. The viscous dissipation of mean-flow energy near the bed makes a leading-order contribution, and an energetic approach to entrainment captures detrainment of fluid through particle settling. These observations allow a reconsideration of particle suspension, advancing over 50 years of research. We find that the new formulation can describe the full evolution of a shallow dilute current, with the accuracy depending primarily on closures for the profiles and source terms. Critically, this enables accurate and computationally efficient hazard risk analysis and earth surface modelling.

The transport of droplets in microfluidic channels is strongly dominated by interfacial properties, which makes it a relevant tool for understanding the mechanisms associated with the presence of more or less soluble surfactants. In this paper, we show that the mobility of an oil droplet pushed by an aqueous carrier phase in a Hele-Shaw cell qualitatively depends on the nature of the surfactants: the drop velocity is an increasing function of the drop radius for highly soluble surfactants, whereas it is a decreasing function for poorly soluble surfactants. These two different behaviours are experimentally observed by using two families of surfactant with a carbon chain of variable length. We first focus on the second regime, observed here for the first time, and we develop a model which takes into account the flux of surfactants on the whole droplet interface, assuming an incompressible surfactant monolayer. This model leads to a quantitative agreement with the experimental data, without any adjustable parameter. We then propose a model for a stress-free interface, i.e. for highly soluble surfactants. In these two limits, the models become independent on the physico-chemical properties of the surfactants, and should be valid for any surfactant complying with the incompressible or stress-free limit. As such, we provide a theoretical framework with two limits for all the experimental physico-chemical configurations, which constitute the bounds for the droplet mobility for intermediate surfactant solubility.

We investigate the momentum fluxes between a turbulent air boundary layer and a growing–breaking wave field by solving the air–water two-phase Navier–Stokes equations through direct numerical simulations. A fully developed turbulent airflow drives the growth of a narrowbanded wave field, whose amplitude increases until reaching breaking conditions. The breaking events result in a loss of wave energy, transferred to the water column, followed by renewed growth under wind forcing. We revisit the momentum flux analysis in a high-wind-speed regime, characterized by the ratio of the friction velocity to wave speed $u_\ast /c$ in the range $[0.3\,{-}\,0.9]$, through the lens of growing–breaking cycles. The total momentum flux across the interface is dominated by pressure, which increases with $u_\ast /c$ during growth and reduces sharply during breaking. Drag reduction during breaking is linked to airflow separation, a sudden acceleration of the flow, an upward shift of the mean streamwise velocity profile and a reduction in Reynolds shear stress. We characterize the reduction of pressure stress and flow acceleration through an aerodynamic drag coefficient by splitting the analysis between growing and breaking stages, treating them as separate subprocesses. While drag increases with $u_\ast /c$ during growth, it decreases during breaking. Averaging over both stages leads to a saturation of the drag coefficient at high $u_\ast /c$, comparable to what is observed at high wind speeds in laboratory and field conditions. Our analysis suggests that this saturation is controlled by breaking dynamics.