We investigate the incompressible flow inside a two-dimensional square cavity, driven by the sliding motion of its four lids, all at the same speed and with facing lids moving in opposite directions. The problem has three symmetries: two mirror symmetries with respect to the diagonals and a $\pi$ rotation invariance about the centre of the cavity. The base flow, a steady state that has all three symmetries, is the unique solution at sufficiently low values of the Reynolds number ($ \textit{Re}$) and acts as a global attractor. At higher $ \textit{Re}$, it has become unstable and shares the phase space with a globally attracting space–time symmetric periodic orbit that, in addition to the rotational invariance, is also invariant under evolution over half a period followed by reflection about either of the diagonals. In between, a wealth of solution branches and intervening bifurcations mediate the transition process. In particular, a pair of steady states that break the mirror symmetries but are mirror-symmetry images of each other regulate the appearance and disappearance of a second space–time symmetric periodic orbit and a pair of asymmetric periodic orbits that are also mirror images of each other. The catalogue of instabilities includes both local (two pitchfork, two Hopf, a saddle-node and a cyclic fold) and global (two heteroclinic and one homoclinic) bifurcations. The sequence of transitions is explained in terms of a one-dimensional path through the parameter space of a codimension-four bifurcation: the double zero bifurcation with Z$_2$ symmetry and degeneracy of the third order terms.

This study investigates necklace-vortex systems forming when a laminar shear-wake, generated by two streams merging at the trailing edge of a splitter plate, interacts with a circular cylinder placed downstream in the wake. Hydrogen-bubble flow visualisations were employed in a water channel capable of producing laminar shear-wake flows. In the absence of the cylinder, oppositely signed vorticity in the shear-wake undergoes mutual annihilation. The introduction of the cylinder interrupts this evolution, promoting off-wall flow separation upstream of the cylinder and vortex roll-up. The study primarily focuses on two non-dimensional parameters, the Reynolds number $ \textit{Re}_m$ and the shear ratio $ \textit{SR}$, and presents a mapping of the observed vortex regimes. Increasing $ \textit{Re}_m$ promotes either the formation of additional vortices or unsteadiness. Increasing $ \textit{SR}$ generally suppresses vortex formation or attenuates unsteadiness, except near $ \textit{SR}\approx 0$ at low to moderate $ \textit{Re}_m$, where the two-vortex system is unstable to additional vortex generation. Observed configurations range from no-vortex states to one- or two-vortex systems at low Reynolds numbers, and to three-, four- and five-vortex systems at larger Reynolds numbers, with unsteadiness becoming prominent beyond the three-vortex regime and predominant in four- and five-vortex systems. Beyond regime mapping, we delve into the structure of a steady two- and three-vortex system at low to moderate $ \textit{Re}_m$. This provides insights into the emergence and evolution of the vortex system, which is analysed in the context of the vorticity-transport equations.

The force on a spherical particle moving arbitrarily in a fairly general class of unsteady potential flows is calculated. It is found that the force, which can be expressed as the gradient of a potential, takes the form of an infinite series of progressively higher order in the ratio of the sphere’s radius to the scales of the surrounding flow. The first few terms conform with the known form usually referred to as ‘added mass force’ but, when the remaining terms are non-negligible, this concept becomes less relevant. Thus, unlike Faxèn’s extension of Stokes’s drag law, it proves impossible to generalise the idea of added mass to particles that are not much smaller than the flow scales. The kinetic energy of the fluid in excess of that which it would have in the absence of the sphere is also calculated when the flows considered are unbounded.

We investigate the transitional flow regimes arising from the interaction between buoyancy and shear in Rayleigh–Bénard–Poiseuille (RBP) flows, considering both large and small domains. The transition boundaries between the bistable system consisting of spiral defect chaos (SDC) and ideal straight rolls (ISRs) in Rayleigh–Bénard convection, and subcritical turbulence in plane Poiseuille flows are not known. Using direct numerical simulations in a large spatial domain over a range of Rayleigh numbers, $Ra \in [0, 10000]$, Reynolds numbers, $\textit{Re} \in [0, 2000]$ and unit Prandtl number, we identify five distinct regimes: (i) bistable SDC and ISRs; (ii) ISRs; (iii) wavy rolls; (iv) intermittent rolls; and (v) shear-driven turbulence. The newly identified intermittent rolls state features longitudinal rolls that decay towards the laminar state before regenerating. In the turbulent regime, longitudinal rolls may coexist with turbulent–laminar bands, highlighting the role of longitudinal rolls in transitional RBP flows. To this end, we examine the unstable manifold of longitudinal rolls in a small domain, integrating along which led to turbulence. This turbulence occur transiently, decaying towards the unstable laminar base state where the longitudinal rolls can be excited again, forming a quasi-cyclic process referred to as the thermally assisted sustaining process (TASP). We further investigate the behaviour of TASP as $\textit{Re}$ and $Ra$ are varied, revealing a stable periodic orbit around the longitudinal roll and the laminar state, and a pathway towards turbulence above a certain $\textit{Re}$ threshold. Finally, we provide a state space sketch of the dynamical processes, emphasising the role of longitudinal rolls in transitional RBP in small domains, and discuss the potential connections to large domains.

Active particles exhibit complex transport dynamics in flows through confined geometries such as channels or pores. In this work, we employ a generalised Taylor dispersion (GTD) theory to study the long-time dispersion behaviour of active Brownian particles in an oscillatory Poiseuille flow within a planar channel. We quantify the time-averaged longitudinal dispersion coefficient as a function of the flow speed, flow oscillation frequency and particle activity. In the weak-activity limit, asymptotic analysis shows that activity can either enhance or hinder the dispersion compared with the passive case. For arbitrary activity levels, we numerically solve the GTD equations and validate the results with Brownian dynamics simulations. We show that the dispersion coefficient can vary non-monotonically with both the flow speed and particle activity. Furthermore, the dispersion coefficient shows an oscillatory behaviour as a function of the flow oscillation frequency, exhibiting distinct minima and maxima at different frequencies. The observed oscillatory dispersion results from the interplay between self-propulsion and oscillatory flow advection – a coupling absent in passive or steady systems. Our results show that time-dependent flows can be used to tune the dispersion of active particles in confinement.

We investigate the emergence of an anomalous solutocapillary instability in an isothermal nanofluid layer with a non-deformable liquid–gas interface. A model of the equation of state for the colloidal suspension is presented. The surface tension exhibits non-monotonic variation with nanoparticle concentration due to nanoparticle surface energetics. In what follows, we consider nanoparticle interfacial kinetics and express the dynamics of surface concentration via the spatio-temporal evolution equation. We analyse the linear stability around the quiescent base state using normal modes and deduce the linear eigenvalue problem to determine the growth rates of these modes. The analytical solution for the monotonic solutocapillary instability is found. Surprisingly, the system displays the onset of an anomalous short-wave solutocapillary instability due to an increase in surface tension with the particle concentration.

We give evidence of non-modal amplification mechanisms driven by swirl intensity from a bi-global linear analysis of a cold swirling flow representative of a premixed swirl burner: non-uniform, compressible, turbulent, enclosed and subject to vortex breakdown passed the expansion. The monolithic computational approach embeds a realistic axisymmetric swirler model in the computational domain. The amplification mechanisms are identified by stability and resolvent analysis under variations of the length of the annular duct section and combustion chamber, the swirl intensity and the swirler position. While the spectrum is affected by changes in the length only, the gain of the resolvent strongly depends on the swirl intensity. The results suggest an acoustically dominated amplification in the combustion chamber and a non-modal hydrodynamic-dominated process driven by the swirl intensity. Inertial waves carrying swirl fluctuations play a key role in the latter. The results are complemented by a resolvent sensitivity analysis that identifies the tip of the inner recirculation region and the surrounding shear layer as a wavemaker region that drives at high swirl numbers the non-modal amplification. The sensitivity of that region also enables the transfer of azimuthal momentum perturbations to axial momentum, hence activating a longitudinal acoustic resonance from azimuthal fluctuations.

Wall-pressure fluctuations beneath turbulent boundary layers (BLs) drive noise and structural fatigue through interactions between fluid and structural modes. Conventional predictive models for the spectrum – such as the widely accepted Goody model (2004 AIAA J., vol. 42 (9), pp. 1788–1794) – fail to capture the energetic growth in the low-frequency range that occurs at high Reynolds number, while at the same time over-predicting the variance. To address these shortcomings, two semi-empirical models are proposed for the wall-pressure spectrum in canonical turbulent BLs, pipes and channels for friction Reynolds numbers $\delta ^+$ ranging from 180 to 47 000. Consistent with the approach outlined modelling the streamwise Reynolds stress in the recent work of Gustenyov et al. (2025 J. Fluid Mech., vol. 1016, A23), the models are based on consideration of two spectral components that represent the contributions to the wall-pressure fluctuations from inner-scale motions and outer-scale motions. The first model expresses the pre-multiplied spectrum as the sum of two overlapping log-normal components: an inner-scaled term that is $\delta ^+$-invariant and an outer-scaled term whose amplitude broadens smoothly with $\delta ^+$. Calibrated against large-eddy simulations, direct numerical simulations and recent high-$\delta ^+$ pipe data, it reproduces the inner-scaled peak and the emergence of an outer-scaled peak at large $\delta ^+$. The second model, developed around newly available pipe data, uses theoretical arguments to prescribe the spectral shapes of the inner and outer components. Embedding the $\delta ^+$-dependence in smooth asymptotic functions yields a formulation that varies continuously with $\delta ^+$ and generalises beyond the calibration range. Both models capture the full spectrum and recover the observed logarithmic growth of its variance, providing a compact, physics-informed empirical representation for more accurate engineering predictions of wall-pressure fluctuations.