We consider two-dimensional irrotational steady gravity waves and derive new explicit analytical bounds for the velocity field and the slope of the free surface. Using an auxiliary function tailored to the streamfunction formulation, we obtain an explicit exponential decay estimate which is optimal for linear waves. The same method yields a new slope estimate that improves existing bounds in the moderate-amplitude regime.

The Atlantic Meridional Overturning Circulation (AMOC), partially driven by double-diffusive horizontal convection (DDHC), plays a key role in regulating the global climate. Indeed, it governs the transfer of heat, salinity and nutrients between the equator and polar regions. The present study investigates an idealised model system, ‘thermohaline circulation in a box’ or ‘AMOC in a box’, namely DDHC in a well-defined geometry, specifically the flow in a box with horizontal temperature and salinity gradients. By varying the temperature Rayleigh number $\textit{Ra}_T$ and the density ratio $\varLambda$, or equivalently the salinity Rayleigh number $\textit{Ra}_S$, four distinct regimes are found. These regimes are distinguished by the global response parameters of the system, namely the temperature Nusselt number $\textit{Nu}_T$, the salinity Nusselt number $\textit{Nu}_S$ and the friction Reynolds number $\textit{Re}_\tau$, as well as by the flow structures. The two limiting regimes of horizontal convection, at high and low $\varLambda$ values, follow the Shishkina-Grossmann-Lohse theory for horizontal convection. In the two regimes in between, in which strong competition between temperature and saline buoyancy occurs, a clear thermohaline layering and the presence of oscillating convected salt fingers are found.

This study investigates a self-similar solution as intermediate asymptotics describing cylindrical shock waves driven by a piston in van der Waals gas under solid-body rotation. The solution is obtained for the exponential variations in the ambient density and the shock radius. The viscous stress follows Newton’s law of viscosity, while heat flux obeys Fourier’s law of heat conduction. The viscosity and thermal conductivity coefficients follow power-law dependencies on temperature and density. The solutions exist with pressure correction for increasing ambient density, while with volume correction for constant ambient density. The viscosity and volume corrections tend to weaken the shock, whereas the pressure correction and the specific heat ratio enhance it. Shock-induced compression intensifies with increasing viscosity, pressure correction and specific heat ratio, but decreases with increasing volume correction. The temperature and density exponents in the viscosity coefficient significantly affect shock compression, shock strength and the distribution of flow variables. Reduced density and radial velocity decrease with viscosity and heat conduction. Viscosity enhances tangential velocity while heat flux affects the normal and tangential stresses. The total energy behind the shock scales as the sixth power of the shock radius for pressure correction and the fourth power for volume correction. Solid body rotation is coupled with shock Mach number and ratio of specific heats. The reduced density, radial velocity and heat flux decrease, while pressure and normal viscous stress increase with pressure correction. Volume correction leads to decreases in density and pressure, but increases in tangential velocity and heat flux.

The time evolution of Beltrami fields in the presence of a time-dependent background flow with spatially homogeneous velocity gradient is analysed using the barotropic vorticity equation. For backgrounds comprising a time-dependent isotropic expansion/contraction and a time-dependent solid-body rotation, we show that every scalar Laplacian eigenfunction generates an unsteady solution of the nonlinear vorticity equation in which the non-background component remains a time-dependent Beltrami field. We derive the evolution law for the background angular velocity in the presence of time-dependent deviatoric strain and velocity divergence, and we generalise the Chandrasekhar–Kendall construction to obtain unsteady Beltrami velocity fields. When the background deformation is a similarity (vanishing deviatoric strain), the Beltrami field is frozen into an advecting flow that differs from the background only by a spatially homogeneous, time-dependent drift. In general, deviatoric strain breaks the Beltrami property, but in regimes where departures are small, we introduce a ‘Beltrami field approximation’. Because the background velocity gradient has nine time-dependent degrees of freedom, three of which are constrained by the vorticity equation, six remaining functions may be prescribed to drive the Beltrami field. We illustrate the approach by describing elastic scattering of Beltrami fields by a background-flow pulse.

Drop tower experiments have been performed to study the capillary collapse of large high-aspect-ratio cavities. Cavities are formed by momentarily impinging the free surface of a liquid bath with a jet of air in the microgravity environment of a drop tower. The collapse may give rise to a jet and three distinct jetting regimes are identified. Simulations are performed to further investigate the phenomena. The abrupt emergence of a thin high velocity jet is observed experimentally and numerically at a specific initial cavity aspect ratio. Different power laws are identified in different regions of the cavity during the collapse providing further understanding of cavity collapse phenomena. In particular, it shows that the spherical and cylindrical self-similar collapses can compete simultaneously, that is, during the same collapse, for determining the final thin jet formation.

In the present study, we propose a novel skin-friction prediction formula based on a re-established self-similarity within the adverse-pressure-gradient (APG) turbulent boundary layer. The basic idea lies in introducing a novel velocity scale, which is derived mathematically and adapted physically from the linear total stress within the boundary layer. This scale assimilates concurrently and fundamentally the friction velocity, two distinct pressure-gradient velocity scales and the half-power law of the mean velocity in the intermediate region. Then, this scale formula is well validated across a comprehensive, multi-geometry database of APG flows over flat plates, curved plates, ramps and airfoils, which covers an unprecedented parameter range, with friction Reynolds number ranging from $10^2$ to $5\times 10^3$ and the Rotta–Clauser pressure-gradient parameter spanning from $10^{-1}$ to $10^2$. Crucially, the proposed scale consistently recovers a classical logarithmic region across all tested APG conditions, thereby restoring the self-similar structures traditionally absent in strong or non-equilibrium pressure-gradient flows. Leveraging this reconstructed self-similarity, we further formulate a new, robust skin-friction prediction model which demonstrates predictive errors confined within $\pm 20\,\%$ for all the investigated non-equilibrium flow states.

The evolution of main shock waves generated by multiple finite-energy blast sources with time-delayed energy release is investigated. We demonstrate that a merged shock wave forms when the individual blasts interact within a critical interval of delay times. This interval is determined principally by the initial sound speed ratio between the compressed gas of the blasts and the ambient medium. Numerical simulations confirm both the existence and the boundaries of this merging regime. Extending the analysis to the strong-shock stage, we derive an asymptotic solution that incorporates an equivalent energy release and a temporal scaling for the dual-source system, thereby describing the propagation of the merged shock front. These results provide a foundational model for multiple blast–shock processes, with potential applications to astrophysical phenomena, intense laser–matter interactions, and blast-wave dynamics in general.

An asymptotic theory is developed to investigate the interaction of surface water waves and a compressible muddy seabed in water of intermediate depth. The water column is treated as inviscid outside a thin bottom boundary layer, the thickness of which is assumed to be of the same order of magnitude as the wave amplitude. The seabed is modelled as an isotropic and homogeneous poro-viscoelastic layer with an incompressible solid skeleton and a compressible pore fluid. The thickness of the seabed is assumed to be comparable to the wave amplitude. Using a perturbation approach, the leading-order analytical solutions for the wave and mud flow are derived, while the evolution of the wave envelope, the mean Eulerian velocity and the mass transport velocity beneath progressive waves are obtained at the second order. Based on the present solution, the wave motion and the induced mass transport are analysed and compared with previous solutions that ignore the compressibility of the seabed. The results demonstrate that the compressibility of the seabed plays a critical role in modulating the flows within both the seabed and the overlying bottom boundary layer. Neglecting compressibility may lead to an underestimation of the interface vertical displacement in highly elastic beds and an overestimation in viscous-dominated cases. Consequently, the Reynolds stress distributions in these regions deviate significantly from predictions based on incompressible seabed theory. This inaccuracy further propagates to the prediction of second-order steady currents and mass transport velocities.

We investigate the onset of transient natural convection in fluid layers subject to volumetric radiative heating and surface cooling. Linear stability analysis reveals a non-monotonic evolution of stability in deep layers, where the flow undergoes successive stages of initial destabilisation, intermediate suppression and eventual restabilisation. This complex temporal behaviour necessitates the definition of dual critical Rayleigh numbers: a lower bound marking the onset of initial instability and an upper bound required for sustained convection. To efficiently predict these thresholds, we develop a local Rayleigh number model that depends solely on the instantaneous conductive temperature profile. When the Rayleigh number exceeds the lower bound, the critical time $t_c$ for flow onset is determined through transient linear stability analysis, and scaling laws are derived to characterise the dependence of $t_c$ on the Rayleigh number $\textit{Ra}$, Prandtl number $\textit{Pr}$, cooling parameter $\phi$ and layer depth $H$. Two distinct instability triggering mechanisms are identified: a top-triggered regime, where $t_c \sim [\textit{Ra} \textit{Pr} \phi /(2+\textit{Pr})]^{-1/2}$, and a bottom-triggered regime, where $t_c \sim [\textit{Ra} \textit{Pr} \exp (-H)/(2+\textit{Pr})]^{-1/2}$. All theoretical predictions are rigorously validated against direct numerical simulations, providing a unified predictive framework for convective onset in systems governed by coupled effects of radiation absorption and surface cooling.

This study investigates the influence of a rear-attached splitter plate on the vortex dynamics and turbulence characteristics in the wake of a circular cylinder. Three-dimensional direct numerical simulations (DNS) are performed at a turbulent Reynolds number of 1000 and non-dimensional plate lengths ($ L/D$) of 0–4. The splitter plate affects the vortex dynamics and turbulence characteristics in a highly non-monotonic manner. Specifically, both the strength of the primary vortices and the turbulent kinetic energy in the wake decrease with increasing $ L/D$ over $ L/D$ = 0–1.5, followed by a local increase over $ L/D$ = 1.5–2 and another decrease over $ L/D$ ≥ 2. The abnormal local increase is because the vortex formation location transitions from downstream of the splitter plate for $ L/D$ ≤ 1.5 to the two sides of the plate for $ L/D$ ≥ 2. Owing to the presence of the primary vortex street in the wake, the turbulence in the wake is strongly anisotropic both globally and locally. After removing the contribution from the coherent primary vortices, the wake becomes much more isotropic globally and homogeneous locally. The present DNS dataset also enables an evaluation of several widely used surrogate models for the kinetic energy dissipation rate. A major finding is that a minor increase in the contribution of the coherent component (e.g. for $ L/D$ ≥ 2) may strongly deteriorate the applicability of the surrogate models of local axisymmetry and local homogeneity. In general, this study provides new insights and mechanisms for the control of turbulence in bluff-body wakes.

The instability of baroclinic Rossby waves over two-dimensional topography is examined using a nonlinear model, a linear stability calculation and wave triads. In all cases, the Rossby waves are unstable, as seen previously over a flat bottom. But topography decreases the growth rates and changes the structure of the unstable waves. When the topographic height (or slope) exceeds a critical value, the instability is ‘locked’ to topography, in that the most unstable mode, particularly in the lower layer, resembles the bathymetry. In this limit, the growth rate becomes independent of topographic height. A triad calculation suggests that the growth rates in the locked state should depend on the lateral scale of the bathymetry but not its height, and that locking does not occur for topographic scales smaller than the surface deformation radius. The results suggest an alternate way that topographically locked flow can be generated, and indicate that baroclinic instability can be much different over steep bathymetry.

Interfacial instability dominates the dynamics as a cavitation bubble oscillates in close proximity to a liquid surface, driving perturbations on both the bubble wall and the liquid surface. The penetration of the liquid layer initiates ventilation, exposing the bubble interior and thereby altering its subsequent dynamics. To quantitatively elucidate the interfacial coupling-induced instability, we develop a theoretical model that couples the perturbation equation with the bubble oscillation equation, considering the liquid viscosity. The model predicts the transition boundaries between ventilation patterns by critical stand-off parameters, which scale exponentially with the liquid viscosity to the −1/3 power. The boundary between complete and partial ventilation regimes shows negligible viscous dependence due to the vanishingly short perturbation growth time. Furthermore, we derive the scaling law of ventilation time, defining it as the instant of perturbation penetration. A series of experiments on bubble oscillation near a liquid surface was conducted, which verified the predictions of the theoretical model. This offers a practical framework for the engineering application of near-surface bubble collapse.