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.

We investigate turbulent flow between two concentric cylinders, oriented either axially or azimuthally. The axial configuration corresponds to a concentric annulus, where curvature is transverse to the flow, while the azimuthal configuration represents a curved channel with longitudinal curvature. Using direct numerical simulations, we examine the effects of both types of curvature on turbulence, varying the inner radius from $r_i=0.025\delta$ to $r_i=95.5\delta$, where $\delta$ is the gap width. The bulk Reynolds number, based on bulk velocity and $\delta$, is set at $R_b\approx 5000$, ensuring fully turbulent conditions. Our results show that transverse curvature, although breaking the symmetry of axial flows, induces limited changes in the flow structure, leading to an increase in friction at the inner wall. In contrast, longitudinal curvature has a significant impact on the structure and statistics of azimuthal flows. For mild to moderate longitudinal curvatures ($r_i\gt 1.5\delta$), the convex wall stabilises the flow, reducing turbulence intensity, wall friction and turbulent kinetic energy (TKE) production. For extreme longitudinal curvatures ($r_i\leqslant 0.25\delta$), spanwise-coherent flow structures develop near the inner wall, leading to a complete redistribution of the TKE budget: production becomes negligible near the inner wall, while pressure–velocity correlations increase substantially. As a result, the mean TKE peaks near the inner wall, thereby weakening the stabilising effect of convex curvature.

We investigate the dynamics and the stability of the incompressible flow past a corrugated dragonfly-inspired airfoil in the two-dimensional (2-D) $\alpha {-}Re$ parameter space, where $\alpha$ is the angle of attack and $Re$ is the Reynolds number. The angle of attack is varied in the range of $-5^{\circ } \leqslant \alpha \leqslant 10^{\circ }$, and $Re$ (based on the free stream velocity and the airfoil chord) is increased up to $Re=6000$. The study relies on linear stability analyses and three-dimensional (3-D) nonlinear direct numerical simulations. For all $\alpha$, the primary instability consists of a Hopf bifurcation towards a periodic regime. The linear stability analysis reveals that two distinct modes drive the flow bifurcation for positive and negative $\alpha$, being characterised by a different frequency and a distinct triggering mechanism. The critical $Re$ decreases as $|\alpha |$ increases, and scales as a power law for large positive/negative $\alpha$. At intermediate $Re$, different limit cycles arise depending on $\alpha$, each one characterised by a distinctive vortex interaction, leading thus to secondary instabilities of different nature. For intermediate positive/negative $\alpha$, vortices are shed from both the top/bottom leading- and trailing-edge shear layers, and the two phenomena are frequency locked. By means of Floquet stability analysis, we show that the secondary instability consists of a 2-D subharmonic bifurcation for large negative $\alpha$, of a 2-D Neimark–Sacker bifurcation for small negative $\alpha$, of a 3-D pitchfork bifurcation for small positive $\alpha$ and of a 3-D subharmonic bifurcation for large positive $\alpha$. The aerodynamic performance of the dragonfly-inspired airfoil is discussed in relation to the different flow regimes emerging in the $\alpha {-}Re$ space of parameters.

Dispersion in spatio-temporal random flows is dominated by the competition between spatial and temporal velocity resets along particle paths. This competition admits a range of normal and anomalous dispersion behaviours characterised by the Kubo number, which compares the relative strength of spatial and temporal velocity resets. To shed light on these behaviours, we develop a Lagrangian stochastic approach for particle motion in spatio-temporally fluctuating flow fields. For space–time separable flows, particle motion is mapped onto a continuous time random walk (CTRW) for steady flow in warped time, which enables the upscaling and prediction of the large-scale dispersion behaviour. For non-separable flows, we measure Lagrangian velocities in terms of a new sampling variable, the average number of velocity transitions (both temporal and spatial) along pathlines, which renders the velocity series Markovian. Based on this, we derive a Lagrangian stochastic model that represents particle motion as a coupled space–time random walk, that is, a CTRW for which the space and time increments are intrinsically coupled. This approach sheds light on the fundamental mechanisms of particle motion in space–time variable flows, and allows for its systematic quantification. Furthermore, these results indicate that alternative strategies for the analysis of Lagrangian velocity data using new sampling variables may facilitate the identification of (hidden) Markov models, and enable the development of reduced-order models for otherwise complex particle dynamics.