At scales larger than the forcing scale, some out-of-equilibrium turbulent systems (such as hydrodynamic turbulence, wave turbulence and nonlinear optics) exhibit a state of statistical equilibrium where energy is equipartitioned among large-scale modes, in line with the Rayleigh–Jeans spectrum. Key open questions now pertain to either the emergence, decay, collapse or other non-stationary evolutions from this state. Here, we experimentally investigate the free decay of large-scale hydroelastic turbulent waves, initially in a regime of statistical equilibrium. Using space- and time-resolved measurements, we show that the total energy of these large-scale tensional waves decays as a power law in time. We derive an energy decay law from the theoretical initial equilibrium spectrum and the linear viscous damping, as no net energy flux is carried. Our prediction then shows a good agreement with experimental data over nearly two decades in time, for various initial effective temperatures of the statistical equilibrium state. We further identify the dissipation mechanism and confirm it experimentally. Our approach could be applied to other decaying turbulence systems, with the large scales initially in statistical equilibrium.

The linear instability of liquid film with insoluble surfactants on a quasiperiodic oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of insoluble surfactants and quasiperiodic oscillation on the instability are described using Floquet theory. For long-wavelength instability, the solution in the limit of long wave perturbations is obtained by the asymptotic expansion method. The results show that a new stable region emerges in the low-frequency domain of the neutral stability curve in the absence of gravity. As the imposed frequency increases, this newly formed stable region is progressively absorbed into a broader stable zone. The U-shaped neutral curves with separation bandwidth appear in the presence of gravity, and the presence of the surfactants will decrease the unstable frequency bandwidth and increase the critical Reynolds number. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. Both travelling-wave and standing-wave modes are found due to the existence of surface surfactants. As the surfactant concentration increases, the finite-wavelength instability region expands significantly, and the intersection point marking the transition from travelling waves to standing waves shifts progressively towards lower frequencies. The physical mechanisms underlying perturbation growth are further elucidated through an energy budget analysis. Energy budget analysis demonstrates that long-wavelength instability is dominated mainly by surface shear stress, whereas finite-wavelength instability is primarily governed by the combined effects of Reynolds stress and surface shear stress.

This paper reports analytical solutions for steadily travelling two-dimensional water waves on deep water, without gravity or surface tension, carrying a cotravelling periodic row of hollow vortices. The solutions are hollow-vortex regularisations of the exact solutions of Crowdy & Roenby (Fluid Dyn. Res., vol. 46, 2014, 031424) for the analogous waves carrying a submerged point-vortex row, the free-surface shapes of which coincide with those for pure capillary waves and, like those, exhibit steady pinchoff at a critical wave amplitude. The same pinchoff phenomenon is shown to occur for the hollow-vortex regularisations. The new wave solutions are likely to provide a useful basis for perturbative, asymptotic or numerical studies when additional effects such as gravity, capillarity or compressibility are incorporated.

A combined experimental and numerical investigation of equilibrium states arising from quasi-two-dimensional turbulent flows in a rotating quadrangular basin with a central flat region and steep slopes adjacent to the sidewalls is presented. The study examines freely decaying and continuously forced regimes. Laboratory experiments show that decaying turbulence consistently evolves into a robust equilibrium state characterised by: (i) a boundary current around the basin along the topographic contours, and (ii) a central anticyclone – features accurately reproduced by shallow-water numerical simulations at laboratory scale. Additional simulations using a mesoscale basin suggest the relevance of these findings to oceanic regimes for different initial conditions and topographic parameters. In the case of continuously forced flows, time-averaged fields reveal qualitatively similar structures, despite the randomness of the applied forcing and the consequent absence of a strict equilibrium. These results demonstrate the emergence of robust flow patterns with implications for the modelling and understanding of semi-permanent flows that are often found in statistical theories of geophysical turbulence.

The wake dynamics of a circular cylinder oscillating in the streamwise direction within a stably (density) stratified fluid is investigated using two-dimensional numerical simulations: Floquet stability analysis and dynamic mode decomposition. At a fixed Reynolds number ($ \textit{Re}=175$) and forcing frequency ratio ($f_d/f_{St}=1.6$), we examine the effects of the oscillation amplitude ($0.1 \leqslant A_D \leqslant 0.6$) and the stratification strength ($1 \leqslant \textit{Fr} \leqslant \infty$) on the wake structure and its symmetry breaking. In unstratified (homogeneous) flow ($ \textit{Fr} = \infty$), the wake transitions from an asymmetric vortex street at low amplitudes to a symmetric state at higher amplitudes. This transition occurs via a Neimark–Sacker bifurcation, with Floquet analysis identifying a critical amplitude of $A_D = 0.455$. In stratified flow, buoyancy forces improve symmetry and suppress vortex shedding for $A_D=0$. At $ \textit{Fr} = 1$, symmetry breaking first occurs at a threshold of $A_D = 0.246$, associated with a period-doubling bifurcation and subharmonic antisymmetric vortex shedding, and persists only within a finite amplitude window ($0.246 \lt A_D \lt 0.560$), beyond which the wake restabilises into a symmetric pattern. At a fixed small amplitude ($A_D = 0.1$), a secondary critical transition is observed at $ \textit{Fr} = 1.52$, marked by quasiperiodic antisymmetric shedding through a near-resonant Neimark–Sacker bifurcation. Stratification also influences force production: moderate stratification ($ \textit{Fr} \approx 2$) minimises drag through enhanced pressure recovery and suppressed wake asymmetry. These results highlight the dual role of stratification in promoting or delaying symmetry-breaking instabilities and modifying wake dynamics. Critical transition thresholds are established, providing insight into buoyancy-modulated flow control strategies relevant to geophysical and engineering applications involving oscillating bodies in stratified environments.

The interaction between marine floating structures and projectiles during water entry plays a crucial role in understanding fluid–structure interactions in polar and offshore environments. This study investigates the impact dynamics of a projectile on a floating structure, emphasising the fluid–structure coupling effects, including the impact-induced cavity evolution, stress wave propagation and fragmentation processes. The computational approach integrates fluid dynamics and discrete element methods (CFD-DEM), allowing for detailed simulation of multi-phase interactions during projectile impact. To address the disparity between fluid grid resolution and particle scale, a dual-grid strategy is incorporated, enabling accurate resolution of multi-scale interactions. The results highlight the fundamental mechanisms of impact water entry, where stress waves radiate through the structure, causing local damage and initiating the formation of fragments. These fragments, in turn, influence the stability of the cavity interface and modify the impact dynamics. The interplay between the floating structure’s buoyant support and the surrounding water contributes to complex load variations on the projectile. Ultimately, the study provides insights into the multi-scale fracture mechanisms induced by projectile impact, with potential applications in improving the design and resilience of structures in dynamic marine environments.

We investigate turbulent Taylor–Couette flow between two concentric cylinders, where the inner cylinder of radius $r_i$ rotates while the outer one of radius $r_o$ remains stationary. Using direct numerical simulations, we examine how varying the radius ratio $\eta = r_i / r_o$ from $\eta = 0.714$ down to $0.0244$ affects the flow characteristics at low to moderate Reynolds numbers. Our results show significant changes in the flow structures and statistics in the limit of a vanishingly small inner radius. The turbulent kinetic energy, scaled with the friction velocity at the inner cylinder, does not exhibit a self-similar scaling; instead, it decreases with decreasing $\eta$. The turbulent kinetic energy budgets reveal that the locations of peak production and total dissipation are independent of $\eta$, whereas their amplitudes decrease as $\eta$ increases. The pressure–velocity correlation near the inner cylinder is large for small $\eta$ and its amplitude decreases with increasing $\eta$, while the turbulent transport term exhibits the opposite trend. Numerical simulations for $\eta \leqslant 0.5$ show that, for our specific set-up, a rather good collapse of the distribution of the normalised torque versus the Taylor number ($ \textit{Ta}$) is obtained when the latter is defined according to Chandrasekhar (Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, 1961), with a tendency towards a $ \textit{Ta}^{1/3}$ regime at sufficiently large $ \textit{Ta}$.

We study the dynamic interaction of two viscous gravity currents in a confined porous layer using laboratory experiments in a vertically placed bead-packed Hele-Shaw cell. By varying the injection rate, along with the density and viscosity of the injecting and ambient fluids, these experiments cover three exact and eight approximate regimes of gravity current interaction, as identified based on the one-dimensional sharp-interface model. By superimposing the theoretically predicted profile shapes and time-dependent frontal locations, a verification is obtained in the different asymptotic regimes of viscous current interaction. Overall, fairly good agreement has been observed between the time-dependent numerical solutions and laboratory measurements on the profile shapes, particularly in the bulk region, where the aspect ratio of the interface shape is fairly large. Such an observation indicates the applicability of the sharp-interface model of viscous current interaction, including the very interesting dynamics of overriding and coflowing. However, the self-similar solutions in some of the exact regimes fail to make reasonable predictions in these experiments, presumably due to the influence of unfinished time transition. We have also observed some degree of disagreement in the frontal regions, which is likely due to the influence of fluid mixing, two-dimensional flow, local heterogeneity and the development of hydrodynamic instabilities for the viscously unstable experiments. The theoretical predictions of the model problem, along with the laboratory experimental observations, offer useful insights into the potential application of, e.g. the technology of co-flooding CO$_2$ and water in oil fields for the co-profits of geological CO$_2$ sequestration and enhanced oil recovery.

This article derives analytical expressions fully describing laminar flow through concentric pipe-within-pipe set-ups, focusing on scenarios where one tube is pressure driven, and the other serves as a lubricant. Both fluid zones are axially unbounded, therefore excluding recirculation, and are connected along longitudinal infinite slits situated on the inner pipe wall, representing fluid–fluid interfaces. Crucially, the viscous interaction along these interfaces is captured by means of a local slip length, for which explicit formulae are provided, allowing a straightforward evaluation. With that, these models provide a full description of the velocity field for slippery concentric pipes, taking into account the viscosity ratio of both fluids and the overall geometry, therefore extending beyond the common assumption of perfect slip applied to superhydrophobic surfaces. Thereby, they enable a precise analysis of the flow, offering important tools to decipher the intricate dynamics of the two coupled fluids within such set-ups. As a result, the insights acquired contribute to the design and optimisation of superhydrophobic and liquid-infused surfaces, with implications for numerous engineering applications such as microfluidic contactors or drag reduction. The analytical models are in excellent agreement with numerical simulations, thus confirming the selected approach. Therefore, our study further illustrates an effective methodology to derive additional analytical models through the presented mathematical techniques, which can serve as a useful template for modelling such surfaces.

The object of investigation in this paper is the nonlinear equations of motion for two-dimensional inviscid water flows with piecewise constant density stratification in a three-layer fluid with a flat bottom, a free surface and two interfaces. We establish a Hamiltonian formulation for the nonlinear governing equations in this set-up. The Hamiltonian of the system and the equations of motion of the surface and of the interfaces are expressed with the help of the Dirichlet–Neumann (DN) operators, which are introduced for each of the layers. Then the linear equations for small amplitudes of the elevation of the surface and of the interfaces in the leading order are derived, from which a bi-cubic equation for the dispersion relation is obtained, whose solutions are analysed. The six real solutions for the possible propagation speeds (three positive, related to right-moving waves, and three negative, related to left-moving waves) have magnitudes of different order. Upper and lower bounds for the previously mentioned roots are also given in terms of the coefficients of the equation. Subsequently, approximate formulas for the propagation speeds are derived. The importance of the DN operators is further illustrated in a separate analysis of the three-layer model with flat surface (rigid lid). The full nonlinear evolution equations are expressed again in terms of the DN operators, and the equations in the linear regime and the weakly nonlinear propagation regime (the Boussinesq approximation) are derived by a proper expansion of the DN operators. Limits to the two-layer free surface model are obtained as well. The obtained results are applicable to internal waves in lakes and in the ocean as well as to laboratory experiments with three superimposed fluid layers.