The equivalent source method (ESM) is one of the fundamental methods for reconstructing the far-field acoustic pressure and identifying the sources in aeroacoustics. However, it suffers from disturbances of near-field hydrodynamic pressure. We propose a hierarchical ESM (HESM) to suppress the disturbance by filtering out hydrodynamic pressure. The hydrodynamic pressure is filtered out using a frequency-domain convection operator. This operator replaces the prescribed convection velocity in Taylor’s frozen flow hypothesis with an adaptive complex convection velocity. The complex convection velocity is adapted in an iterative way to take into account the multiscale convection and spatial amplification of hydrodynamic pressure. The hydrodynamic pressure associated reconstruction error can thus be suppressed. The proposed HESM is compared with the Ffowcs Williams–Hawkings analogy method and the ESM that directly uses the pressure and the filtered pressure with the widely used uniform convection velocity. The use of complex convection velocity-based hydrodynamic pressure filtering mitigates the overprediction of acoustic pressure and enables precise reconstruction of both acoustic pressure directivity and spectra.

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.

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.

We present three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection in a closed rectangular box whose width $L_y$ and length $L_x$ are 0.8 and 2.4 times the height $H$, respectively. The Rayleigh number $\textit{Ra}$ varies from $10^5$ to $10^{10}$, and the Prandtl number is unity. The advantages of the present configuration are: (a) a relatively stable unidirectional large-scale circulation, consisting of two counter-rotating rolls, fills the cell and fixes the thermal plume ejection- and shear-dominated regions, in contrast to those in closed cylindrical cells. (b) The regions of plume ejection are essentially independent of the sidewalls so that their autonomous existence can be studied. This is because there is some space, or ‘fetch’, for the velocity and thermal boundary layers to develop along the length. (c) This geometry allows one to study the influence of locally thin and thick boundary layers (which follow larger or smaller plume activity) on the scaling of convection properties. In regions of larger plume activity (defined by an incessant movement of plumes), the temperature fluctuation as well as the normalised thermal and viscous dissipation rates decay more slowly with $\textit{Ra}$ than in regions of lower activity. Both viscous and thermal boundary layers thin down rapidly with increasing distance from the plume ejection region. The local thicknesses of both boundary layers decline more rapidly with $\textit{Ra}$ in the ejection region than in regions of impact and shear, where they are similar to each other. Despite these details, the global heat transport laws are practically the same as those in other configurations of low to moderate aspect ratios.

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.

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.

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.