Building on classical oblique jump theory, we develop a one-dimensional (1-D) analytical framework that incorporates non-Newtonian rheology to predict the onset of hydraulic jumps, their internal structure and the associated Mach-front geometry. Source terms representing bed slope and wall friction are included, and the resulting formulation is systematically assessed against laboratory experiments, two-dimensional (2-D) shallow-water simulations and fully three-dimensional (3-D) computational fluid dynamics. Experiments with Newtonian, shear-thinning and shear-thickening fluids on converging sidewalls demonstrate a good match with the 1-D formulation. For Newtonian and shear-thinning fluids on mild slopes, the 1-D formulation with source terms closely reproduces the measured shock-front geometry and the 2-D simulation results. The analysis shows that upstream flow deceleration governs the reduction of the Mach angle and the resulting curvature. By contrast, in tests with shear-thickening fluids and steeper slopes, gravitational contributions produce detachment and strong front curvature that are not captured by the 1-D model. Comparisons of the transverse front position confirm that 1-D models lose validity when the upstream Froude number decreases sharply along the front. Fully 3-D simulations reveal concave front deformation driven by shear, strong dominance of tangential over normal velocities and flow features absent in depth-averaged models. The results demonstrate that 2-D shallow-water models capture the key dynamics for mild slopes and shear-thinning conditions, while accurate prediction for shear-thickening fluids requires 3-D approaches, motivating future hybrid strategies.

Torsional vibration and galloping of a triangular prism (TP) in steady flow is investigated numerically at mass ratio 2.5, low Reynolds number 150, three angles of attack, and reduced velocities up to 40. The vibration of the TP is torsional galloping characterised by monotonic increase of the angular amplitude with the increase of reduced velocity. The angular displacement and amplitude are non-dimensionalised by 2π/3, which is the geometrical period in the rotation direction. The response of the TP is well correlated to the direction of the fluid moment coefficient on a stationary TP with a constant rotation angle. The rotation angles are consistently divided into excitation and damping ranges where the directions of the mean fluid moment of a stationary TP and the rotational angle are the same, and opposite to each other, respectively. When the reduced velocity is less than a critical value, the vibration amplitude falls into a damping range, and it increases with the increase of reduced velocity. When the reduced velocity is greater than this critical value, the galloping of the TP is strong and very aperiodic. The vibration amplitude switches very frequently between multiple amplitudes. Every identified amplitude is very close to the upper boundary of a damping range. Multiple-amplitude torsional galloping is a distinct feature that was not found in transverse galloping in the crossflow direction.

This work presents a predictive framework for energy harvesting of two tandem and staggered flapping foils based on four canonical modes of vortex–foil interaction. The role of the incoming vortex generated by the leading foil in modulating the hydrodynamic load of the trailing foil is systematically analysed. Four canonical interaction modes are classified by the vortex rotation and its interaction position relative to the leading-edge vortex (LEV). The most effective configuration occurs when the foil encounters a counter-rotating vortex on the pressure side, which strengthens the LEV and consequently enhances the lift magnitude, with maximum efficiency achieved when vortex merging occurs near stroke reversal. A second constructive mode occurs when a co-rotating vortex on the suction side promotes LEV roll-up through favourable induced velocities. Force decomposition reveals that in both constructive modes, the incoming vortices improve the efficiency of the trailing foil by enhancing the unsteady lift through altering the local velocity to strengthen the LEV or promote its roll-up, while their low-pressure cores contribute marginally to the unsteady force. Two destructive modes are also observed: direct interaction of a counter-rotating vortex on the suction side leads to only a transient lift increase; a co-rotating vortex on the pressure side reduces the effective angle of attack and leads to the poorest performance. Building on these insights, a mechanism-based predictive framework is established to rapidly identify high-performance configurations without exhaustive parametric exploration. The framework applies broadly to different wake conditions and trailing-foil kinematics and guides the design of multi-foil energy-harvesting systems.

Görtler vortices developing over a concave wall support rapidly oscillating wavelike disturbances through secondary instabilities. Although experiments indicate that the finite-amplitude evolution of these waves acts as a precursor to turbulence transition, accurate and efficient prediction has remained out of reach. We overcome this limitation by using the parabolised coherent structures (PCS) method of Song & Deguchi (2025 J. Fluid Mech., vol. 1025, A42), which incorporates the nonlinear vortex-wave interaction into a standard spatial-marching approach. Our computational results agree well with the wave amplitude and displacement thickness observed in the widely known experiments of Swearingen & Blackwelder (1987 J. Fluid Mech., vol. 182, pp. 255–290).

The incipient cavitation of a pair of unequal strength counter-rotating vortices undergoing the long-wavelength Crow instability is examined with high-speed video, acoustic measurements and volumetric particle tracking velocimetry. This work expands upon the previous studies of Chang & Ceccio (J. Acoust. Soc. Am., vol. 130, 2011, pp. 3209–3219) and Chang et al. (Phys. Fluids, vol. 24, 2012, 014107). Volumetric velocimetry results presented by Knister et al. (J. Fluid Mech., 2026) were used to predict the core pressures of the stretched secondary vortices. These data are combined with free-stream nuclei measurements to predict the rates of cavitation inception, which compared well with the directly measured inception rates. The acoustic emissions of incipient cavitation events are also related to the vortex properties and the nuclei content of the water. The reduced pressure in the stretched vortices is shown to be related primarily to the reduction in the core radius of the secondary vortex and not due to axial jetting or straining. The measured vortex dynamics indicates that the process leading to the pressure drop in the secondary vortex core is a transient process but not more rapid than the development of the Crow instability. In conclusion, these results show that a relatively simple model of cavitation inception in a stretched secondary vortex captures the essential physics connecting the nuclei population and the underlying vortical flow field, enabling prediction of the resulting observed inception rates. These results also indicate that the reduced pressures in vortical flow leading to inception are primarily due to the reduction in the core of the vortices and not due to substantive axial jetting. The pressure drop in the vortex cores is accordingly a transient process, but despite the appearances of cavitation inception it does not proceed faster than the development of the Crow instability in the secondary vortices.

The physics of settling suspensions under shear are investigated by theoretical and numerical analyses of unstable equilibrium solutions to the incompressible Navier–Stokes equations, coupled with an advection–diffusion–settling equation for a dilute phase of particles. Two cases are considered: the ‘passive scalar’ regime, in which the sediment is advected by the fluid motion, but concentrations are too dilute to affect the flow; and the ‘stratified’ regime, where a non-uniform vertical distribution of sediment due to particle settling leads to a bulk stratification that feeds back on the flow via buoyancy. In the passive regime, we characterise the structure of the resultant sediment concentration fields and derive formulae for transport fluxes of sediment with asymptotically low and high settling velocities. In the stratified regime, parametric continuation is employed to explore the dependence of states upon the bulk Richardson number $ \textit{Ri}_b$. Symmetry breaking in the governing equations leads to travelling wave solutions with a rich bifurcation structure. The maximum $ \textit{Ri}_b$ attained by these states depends non-monotonically on settling velocity and obeys asymptotic scalings that have also been observed to capture the dependence of the laminar–turbulent boundary in direct numerical simulations.

Fluid pumping in a horizontal slot using the pattern interaction effect has been analysed. This pumping is of interest as it operates without external energy sources beyond those required to create the necessary heating patterns. Activation of this effect involves a combination of fixed surface topography and adjustable heating patterns. The flow rate, including its direction, can be controlled by moving the heating pattern relative to the groove pattern. This analysis extends that of Abtahi & Floryan (2017 J. Fluid Mech. vol. 826, pp. 553–582), who considered only small-amplitude grooves in which the achieved flow rate is proportional to the groove amplitude. Grooves with arbitrary amplitudes spanning the slot were considered in the current analysis, and their most effective heights and distributions have been identified. A detailed analysis was conducted of groove and heating patterns described by a single Fourier mode applied to one or both plates bounding the slot. In all cases, the flow rate increased proportionally to the groove amplitude until an excessively large amplitude caused flow choking, and to the heating intensity until saturation was reached. The groove wavenumber of approximately 0.8 was found to be the most effective in the case of one groove plate, and 0.5–0.7 in the case of two groove plates. The flow rate decreases rapidly at both smaller and larger wavenumbers. The largest flow rate was achieved by placing grooves on both plates to form a wavy slot, with hot spots positioned halfway between the groove peaks and troughs.

Attempts to disentangle shear-flow turbulence often focus on identifying relatively simple solutions, such as travelling waves or periodic orbits. We show, however, that capturing multiscale features requires considering states at least as complex as quasi-time-periodic solutions. Approximations of these states can be computed efficiently using a quasi-linear model, consistent with the large-Reynolds-number asymptotic analysis. The quasi-linear structure is key to producing multiscale critical layers that generate vortices obeying Taylor’s frozen-flow hypothesis.

Accurate prediction of the hydrodynamic coefficients of non-spherical particles in wall-confined flows is crucial for understanding particle–fluid interactions and reliable modelling of particle motion. Under strong wall confinement, the hydrodynamic coefficients exhibit a highly nonlinear dependence on the Reynolds number, wall distance and particle orientation – posing significant modelling challenges. In this study, we propose a multi-stage physics-informed machine-learning (MSPIML) framework for modelling the drag, lift and pitching torque coefficients of a wall-bounded prolate spheroid over the explored parameter space. In the first stage, a physics-informed mixture-of-experts (PIMoE) model predicts the drag coefficient by intelligently blending empirical correlations with a data-driven statistical expert. The resulting high-fidelity drag coefficient is then injected as an auxiliary input to a second-stage model, either a deep neural network (DNN) or an additional MoE, that predicts lift and pitching torque coefficients, thereby leveraging the strong physical coupling among the three coefficients. Trained on a comprehensive dataset of 720 direct numerical simulations covering wide ranges of Reynolds number, wall distance and particle orientation, the optimal PIMoE–DNN and PIMoE–MoE configurations achieve relative errors below 2.2 % for drag, 11.4 % for lift and 7.0 % for pitching torque while maintaining excellent generalisation across the entire parameter space. Moreover, the Shapley additive explanations analysis confirms that the MSPIML framework correctly captures the physical dependencies: dominant influence of Reynolds number and strong pitching torque dependence on the drag coefficient. The MSPIML framework provides an interpretable and efficient approach to the prediction of hydrodynamic coefficients and offers substantial potential for dynamic modelling of non-spherical particles in multiphase flows.

We investigate the stability of the flow past two side-by-side square cylinders (at Reynolds number 200 and gap ratio 1) using tools from dynamical systems theory. The flow is highly irregular due to the complex interaction between the flapping jet emanating from the gap and the vortices shed in the wake. We first perform spectral proper orthogonal decomposition (SPOD) to understand the flow characteristics. We then conduct Lyapunov stability analysis by linearising the Navier–Stokes equations around the irregular base flow and find that it has two positive Lyapunov exponents. The covariant Lyapunov vectors (CLVs) are also computed. Contours of the time-averaged CLVs reveal that the footprint of the leading CLV is in the near-wake, whereas the other CLVs peak further downstream, indicating distinct regions of instability. SPOD of the two unstable CLVs is then employed to extract the dominant coherent structures and oscillation frequencies in the tangent space. For the leading CLV, the two dominant frequencies match closely with the prevalent frequencies in the drag coefficient spectrum and correspond to instabilities due to vortex shedding and jet-flapping. The second unstable CLV captures the subharmonic instability of the shedding frequency. Global linear stability analysis (GLSA) of the time-averaged flow identifies a neutral eigenmode that resembles the leading SPOD mode of the first CLV, with a very similar structure and frequency. However, while GLSA predicts neutrality, Lyapunov analysis reveals that this direction is unstable, exposing the inherent limitations of the GLSA when applied to chaotic flows.

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.

A mathematical model for the deposition of particles from a thin sessile droplet undergoing diffusion-limited evaporation in four different modes of evaporation, namely the constant contact radius (CR), constant contact angle (CA), stick–slide (SS), and stick–jump (SJ) modes, is formulated and analysed. Explicit expressions are obtained for the flow and concentration of particles within the droplet, as well as the evolutions of the mass of particles in the bulk of the droplet and in a distributed deposit and/or in one or more ring deposits on the substrate. It is shown that the nature of the deposit depends on both the local evaporative flux and the motion of the contact line. In particular, for a droplet undergoing diffusion-limited evaporation, the flow is outwards towards the contact line in both the CR and CA modes, however, the receding contact line in the CA mode results in a qualitatively different deposit from that in the CR mode, specifically a switch from a ring deposit in the CR mode to a near-uniform deposit in the CA mode. This contrasts with the behaviour of a droplet undergoing spatially uniform evaporation in the CA mode, in which the flow is radially inwards resulting in a peak deposit. For a droplet evaporating in the SS or SJ modes, the final deposit is a combination of the deposit types associated with the CR and CA modes. The present model is validated by finding good agreement between the theoretical predictions for the deposit and previous experimental results.

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.

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.

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.

The wake systems of a conventional ducted propeller ($\mathrm{DP}$) and a rim driven thruster ($\mathrm{RDT}$) are compared. The latter is an innovative ducted propeller, whose blades are installed on a rim rotating within the nozzle, with their tips oriented inwards and no need of a rotating hub. The flow was reproduced by large eddy simulation (LES) on a cylindrical grid consisting of 6.3 billion points. Substantial deviations between the flow physics downstream of the two propellers are revealed and an order of magnitude drop in the pressure minima and turbulent stresses is found across the rotor and in the wake of $\mathrm{RDT}$. These changes are mainly attributable to the absence of the hub vortex, the helical vortices from the root of the blades, and the leakage flow generated between their tip and the inner surface of the nozzle. In the rim driven thruster, they are replaced by inner, helical vortices shed from the tip of its blades. In addition, the trailing wake of the blades of $\mathrm{RDT}$ is populated by smaller streamwise vortices and lower turbulence levels. This is due to their modified design, characterised by a more uniform spanwise distribution of the load, allowed by the absence of both the hub and the leakage between the tip of the blades and the nozzle. In conventional ducted propellers, they require a reduction of the load of the blades towards their root and their tip with the purpose of mitigating the intensity of the hub vortex and the leakage flow, respectively.