Equilibrium shapes of hollow vortices with surface tension in a corner geometry are obtained by solving a free-boundary problem. Using the integral hodograph method, we derive the complex velocity potential in an auxiliary parameter plane, which includes the velocity magnitude along the free surface. A singular integral equation for the velocity magnitude is obtained by applying the dynamic boundary condition. Numerical solutions to this equation reveal a wave quantisation phenomenon on the boundary of the hollow vortex due to the surface tension. The number of waves allocated on the free surface is arbitrary, starting from some minimal value depending on the strain-to-circulation ratio, the corner angle and the surface tension. In the limiting case of zero surface tension, the solution is obtained analytically and shown to agree with previous studies based on alternative mathematical formulations. These findings provide the first known equilibrium configurations of hollow vortices with surface tension in the presence of solid boundaries.

We use direct numerical simulations to investigate fluid–solid interactions in suspensions of rigid fibres settling under gravity in a quiescent fluid. The solid-to-fluid density ratio is $\mathcal{O}(100)$, while the Galileo number ($ \textit{Ga}$) and fibre concentration ($n\ell_{\kern-1.5pt f}^3$) are varied over the ranges $ \textit{Ga} \in [180, 900]$ and $n\ell_{\kern-1.5pt f}^3 \in [0.36, 23.15]$; $\ell_{\kern-1.5pt f}$ denotes the fibre length and $n$ the number density. At high $ \textit{Ga}$ and/or low $n\ell_{\kern-1.5pt f}^3$, fibres cluster into gravity-aligned streamers with elevated concentrations and enhanced settling velocities, disrupting the flow homogeneity. As $ \textit{Ga}$ increases and/or $n\ell_{\kern-1.5pt f}^3$ decreases, the fluid-phase kinetic energy rises and the energy spectrum broadens, reflecting enhanced small-scale activity. The flow anisotropy is assessed by decomposing the energy spectrum into components aligned with and transverse to gravity. Vertical fluctuations are primarily driven by fluid–solid interactions, while transverse ones are maintained by pressure–strain effects that promote isotropy. With increasing $ \textit{Ga}$, nonlinear interactions become more prominent, producing a net forward energy cascade toward smaller scales, punctuated by localised backscatter events. Analysis of the local velocity gradient tensor reveals distinct flow topologies: at low $ \textit{Ga}$, the flow is dominated by axisymmetric compression and two-dimensional straining; at high $ \textit{Ga}$, regions of high fibre concentration are governed by two-dimensional strain, while voids are associated with axisymmetric extension. The fluid motion is predominantly extensional rather than rotational.

Low Reynolds number hydrodynamic interactions are generally considered both deterministic and reversible due to their linearity. However, the role of soft interactions in deformable suspensions drives nonlinear effects with ambiguous consequences. On the one hand, nonlinearities can be responsible for soft chaos, i.e. long-time apparent randomisation resulting from sensitivity to initial conditions. On the other hand, they can also drive steady streaming and/or drifting effects leading to alignment and ordering. Here, we conduct a comprehensive study on the binary interaction of elastic capsules positioned in different shear planes using high-fidelity particle-resolved simulations. The effects of alignment angle, inter-surface distance, capillary number and size ratio are systematically explored. Based on interaction stability, three regimes are identified: leapfrog, minuet and a novel capturing regime. Unlike leapfrog and minuet motions, where the satellite capsule ultimately escapes from the reference capsule, the capturing motion forms a stable doublet aligned along the vorticity direction. We reveal that capturing is a gentle interaction, which induces only minimal deformation and stress. The mechanism underlying the capturing regime is attributed to the interplay between periodic oscillations induced by the central capsule and steady drift along the vorticity direction. Harmonic analysis of interaction frequencies further underscores the nonlinearity inherent to this dynamics. Extending beyond binary systems, we show that this mechanism relays into ternary alignment, suggesting a generic route to chain formation, demonstrating that nonlinear hydrodynamic interactions alone can drive spontaneous ordering of deformable particles.

The energy of fluid turbulence is transported, on average, to smaller and larger scales in three-dimensional and two-dimensional flows, respectively. The motion along the flat free surface of a turbulent liquid shares similarities with both classes of flows, therefore the direction of the energy cascade along it is ambiguous. We show experimentally that the process is linked to the local divergence of the surface velocity field: expansive motions, associated with flow upwelling towards the surface, transfer energy to larger scales, while compressive motions, associated with fluid plunging into the bulk, do the opposite. The net inter-scale energy flux is therefore vanishingly small, in stark contrast with homogeneous turbulence in both two- and three-dimensional systems. Moreover, we find that rare and intense compressive/expansive events are chiefly responsible for the instantaneous inter-scale fluxes, which are much stronger than their counterparts at depth.

We study the combined effects of natural convection and rotation on the dissolution of a solute in a solvent-filled circular cylinder. The density of the fluid increases with increasing concentration of the dissolved solute, and we model this using the Oberbeck–Boussinesq approximation. The underlying moving-boundary problem has been modelled by combining the Navier–Stokes equations with the advection–diffusion equation and a Stefan condition for the evolving solute–fluid interface. We use highly resolved numerical simulations to investigate the flow regimes, dissolution rates and mixing of the dissolved solute for $Sc = 1$, $Ra \in [10^5, 10^8]$ and $\varOmega \in [0, 2.5]$. In the absence of rotation and buoyancy, the distance of the interface from its initial position follows a square root relationship with time ($r_d \propto \sqrt {t}$), which ceases to exist at a later time due to the finite-size effect of the liquid domain. We then explore the rotation parameter, considering a range of rotation frequency – from smaller to larger, relative to the inverse of the buoyancy-induced time scale – and Rayleigh number. We show that the area of the dissolved solute varies nonlinearly with time depending on $Ra$ and $\varOmega$. The symmetry breaking of the interface is best described in terms of $Ra/\varOmega ^2$.

This paper investigates the aerodynamic and flow characteristics of a circular cylinder near the leading-edge separated flow of an elongated rectangular cylinder. The study varies the gap-to-diameter ratio (G/D) of 0 ≤ G/D ≤ 0.4 and distance-to-diameter ratio (L / D) of 0.6 ≤ L / D ≤ 5.8 in the subcritical Reynolds-number region. Here, D, G and L are the diameter of the circular cylinder, the gap between the two isomeric cylinders and the distance between the leading edge of the rectangular cylinder and the centre of the circular cylinder, respectively. Based on smoke-wire flow visualisations, particle image velocimetry test results, lift power spectral densities and pressure distributions, flow around the circular cylinder can be classified into three regimes, i.e. broadened body, body reattachment and co-shedding. In the broadened-body regime, gap flow is negligible, and the circular cylinder behaves as an extension of the rectangular cylinder. In the body-reattachment regime, the free shear layer separated from the rectangular cylinder’s leading edge reattaches to the circular cylinder forebody, significantly modifying its incoming flow. In the co-shedding regime, the free shear layer substantially alters the vortex shedding from the circular cylinder’s lower side, resulting in a distorted alternating vortex shedding from the circular cylinder. Both the drag and lift of the circular cylinder display distinct behaviours in the three flow regimes. Two primary flow modes are recognised through proper orthogonal decomposition analysis: an alternating vortex shedding mode and a one-sided shear flow mode, which result in two Strouhal numbers of 0.205 and 0.255, respectively.

Roll patterns on floating ice shelves have been suggested to arise from viscous buckling under compressive stresses. A model of this process is explored, allowing for a power-law fluid rheology for ice. Linear stability theory of uniformly compressing base flows confirms that buckling modes can be unstable over a range of intermediate wavelengths when gravity does not play a dominant role. The rate of compression of the base flow, however, ensures that linear perturbations have wavelengths that continually shorten with time. As a consequence, linear instability only ever arises over a certain window of time $t$, and its strength can be characterised by finding the net amplification factor a buckling mode acquires for $t\to \infty$, beginning from a given initial wavenumber. Bi-axial compression, in which sideways straining flow is introduced to prevent the thickening of the base flow, is found to be more unstable than purely two-dimensional (or uni-axial) compression. Shear-thinning enhances the degree of instability in both uni-axial and bi-axial flow. The implications of the theoretical results for the glaciological problem are discussed.

The compression waves/boundary layer interaction (CWsBLI) in high-speed inlets poses significant challenges for predicting flow separation, rendering traditional shock wave/boundary layer interaction (SWBLI) scaling laws inadequate due to unaccounted effects of the coverage range of compression waves. This study aims to establish a unified scaling framework for CWsBLIs and SWBLIs by proposing an equivalent interaction intensity. Experiments were conducted in a Mach 2.5 supersonic wind tunnel, employing schlieren imaging and pressure measurements to characterise flows induced by curved surfaces at two deflection angles ($10^{\circ }, 12^{\circ }$) and varying coverage ranges of compression waves ($d$). An equivalent transformation method was developed to convert the CWsBLI into an equivalent incident SWBLI (ISWBLI), with interaction intensity derived from pressure gradients considering the coverage range. Key results reveal a critical threshold based on the interaction length of ISWBLI ($L_{\textit{single}}$): when $d \leq L_{\textit{single}}$, the interaction scale remains comparable to ISWBLI; when $d \gt L_{\textit{single}}$, the weakened adverse pressure gradient leads to a reduction in the length scale. The proposed scaling framework unifies the CWsBLIs and SWBLIs, achieving better data collapse compared to the existing methods. This work advances our understanding of complex waves/boundary layer interactions, and provides a prediction method for the length scales of CWsBLIs.

Using linear stability analysis, we study the onset and formation mechanism of wall modes in confined magnetoconvection cells with the degree of confinement characterised by the cell aspect ratio $\varGamma$. We first outline the phase diagram of the dominating factors that determine the critical Rayleigh number $Ra_c$ for the onset of convection in the $\varGamma -Ha$ phase space, with $\textit{Ha}$ being the Hartmann number. Our study shows that $Ra_c$ is primarily determined by geometrical confinement, and bulk convection onset occurs with $Ra_c = 1090 \varGamma ^{-4.0}$ for $\varGamma \lt \varGamma _{c_1} = 1.21 \textit{Ha}^{-0.48}$. No wall modes form and $Ra_c$ depends on the strength of both the confinement and magnetic field for $\varGamma _{c_1} \leqslant \varGamma \lt \varGamma _{c_2} = 4.07 \textit{Ha}^{-0.53}$. For $\varGamma _{c_2}\leqslant \varGamma \lt \varGamma _{c_3}=0.99 \textit{Ha}^{-0.10}$, wall modes emerge and $Ra_c$ drops below the bulk onset Rayleigh number for magnetoconvection. When $\varGamma \geqslant \varGamma _{c_3}$, wall modes become fully developed with an onset Rayleigh number for wall modes $Ra_{c,w} \approx 65 \textit{Ha}^{1.5}$. In this fully developed regime, the radial velocity profile and $Ra_{c,w}$ become independent of $\varGamma$. Through analysing the length scales of wall modes and their interaction with spatial confinement, we show dynamically how wall modes emerge in confined cells: while the first layer with a characteristic length scale $\ell _1 = 1.04 \textit{Ha}^{-0.56}$ forms when $\varGamma \geqslant 5.39 \textit{Ha}^{-0.58}$, the second layer with a characteristic length scale $\ell _2 = 4.94 \textit{Ha}^{-0.56}$ emerges when $\varGamma \geqslant 9.07 \textit{Ha}^{-0.53}$. These scaling relations provide practical guidelines for experimental and numerical studies of the wall-mode dynamics.

This study experimentally investigates passive drag reduction on a sphere using azimuthally spaced surface protrusions under subcritical Reynolds numbers, focusing on the effects of the protrusion number at fixed surface coverage. The proposed surface modification strategy, termed partial protrusions, maintains a constant total protruded area while varying the number of protrusions $N$, thereby adjusting their azimuthal spacing. The objective is to determine whether such configurations can outperform the conventional full protrusion, in which protrusions continuously surround the azimuthal direction, and to elucidate the flow mechanisms behind any observed enhancement. Drag and flow field measurements reveal that increasing $N$ significantly improves aerodynamic performance. When $N$ exceeds a certain threshold, the partial-protrusion configuration achieves a greater drag reduction than the full-protrusion case, despite using only half the surface coverage. For low $N$, asymmetric pressure distributions across the protruded and smoothed sides induce unsteady separation delay, leading to shear-layer oscillations and elevated turbulent kinetic energy. As $N$ increases, the azimuthal spacing between protrusions decreases, promoting stable interaction between the two sides and leading to separation delay farther downstream than in the full-protrusion case, along with suppression of flow unsteadiness. These results demonstrate that a well-designed partial-protrusion configuration can outperform the full-protrusion configuration in drag reduction and unsteadiness control, offering new insights into effective passive flow control strategies for bluff body flows.

We investigate the unsteady lift response of compliant membrane wings in hovering kinematics by combining analytical inviscid theory with experimental results. An unsteady aerodynamic model is derived for a compliant thin aerofoil immersed in incompressible inviscid flow of variable free-stream velocity at high angles of attack. The model, representing a spanwise section of a hovering membrane wing, assumes small membrane deformation and attached flow. These assumptions are supported by experiments showing that passive membrane deformation suppresses flow separation when hovering at angles of attack up to $55^\circ$. An analytically derived expression is obtained for the unsteady lift response, incorporating the classical Wagner and Theodorsen functions and the membrane dynamic response. This theoretical expression is validated against experimental water-tank measurements that are performed on hovering membrane wings at angles of attack of $35^\circ$ and $55^\circ$. Data from membrane deformation measurements is applied to the theoretical lift expression, providing the theoretical lift response prediction for each of the available experimental scenarios. Results of the comparison show that the proposed theory accurately predicts unsteady lift contributions from membrane deformation at high angles of attack, provided the deformation remains small and the flow is attached. This agreement between inviscid theory and experimental measurements suggests that when flow separation is suppressed, the unsteady aerodynamic theory is valid well beyond the typical low-angle-of-attack regime.

Smooth surface features were recently found to stabilise stationary cross-flow instability (CFI) of swept-wing boundary layers, thus holding potential for passive laminar flow control. Notably, the effect of surface features on the transition location exhibited a significant dependence on the CFI amplitude. In this work, numerical solutions of the harmonic Navier–Stokes (HNS) equations are used to explore the impact of a smooth surface hump on the linear and nonlinear development of stationary CFI under various perturbation amplitudes. Linear simulations identify regions of successive inhibited and enhanced perturbation growth. Despite the recovery of the base flow and perturbation kinetic energy to the reference (i.e. no-hump) state, significantly reduced perturbation growth is observed. The distorted perturbation profile due to the interaction with the hump is postulated to be responsible for this. Increasing the perturbation amplitude results in a response of the flow that is qualitatively similar to the linear case, albeit with increasing local destabilisation of new fundamental (i.e. primary wavelength) structures and higher-order harmonics near the wall. An energy budget analysis reveals that the growth of the fundamental incoming CFI is inhibited through the reduced effectiveness of the lift-up mechanism downstream of the hump. This is preceded by a spatial perturbation shape deformation, governed by (spanwise) transport terms. The results suggest that stabilisation of incoming stationary CFI via smooth surface humps is most effective at low incoming perturbation amplitudes. At higher perturbation amplitudes, newly formed near-wall structures, pre-conditioned by the incoming CFI, overtake the incoming CFI and could anticipate the transition process.

Fluids at supercritical pressure (SCP) exhibit significant real-fluid effects across the pseudo-critical point, which challenges the validity of the existing wall-scaling laws developed under atmospheric pressure condition. This study revisits prior efforts on the temperature-based transformation for the collapse of mean scalar profiles, emphasising the difficulties in accurately describing universal characteristics of thermal boundary layers at SCP. To address this, a novel thermal scaling law using enthalpy transformation is proposed by incorporating the chain rule and heat flux balance. This transformation effectively accounts for variations in the near-wall thermophysical properties associated with the scalar profile while excluding the gradient of isobaric specific heat capacity-related terms. The proposed scaling law demonstrates substantially improved alignment of transformed mean scalar profiles in SCP channel flows at different wall-temperature differences and Reynolds numbers. Additionally, the enthalpy transformation shows superior performance compared with the existing enthalpy–velocity relations, particularly near the heated-wall region where the fluid thermodynamic states undergo the pseudo-boiling process. The present work could facilitate the development of universal wall model in supercritical flows, enabling rapid and reliable heat transfer predictions in practical applications.

Direct numerical simulation is performed to study the effects of spanwise curvature on transitioning and turbulent boundary layers. Turbulent transition is induced with an array of resolved cuboids. Spanwise curvature is prescribed using a novel approach with a body force that is applied orthogonally to the bulk flow to curve the mean free-stream streamlines at a set radius. The flows are analysed in a streamline-aligned coordinate system. Although the radius of curvature is large compared with the size of the boundary layer, its effects on the development of the boundary layer are appreciable. The results indicate that spanwise curvature induces a non-uniform mean secondary flow and alters the structure of turbulence within the boundary layer. Analytical expressions for the crossflow are derived in the viscous sublayer and log layer. These alterations are visible as changes in the distribution of the turbulent stresses and alignment of the vortical structures with the mean flow. These modifications are responsible for a misalignment between the Reynolds stress tensor and the velocity gradient tensor, which has important consequences for the validity of the widely used Boussinesq turbulent viscosity hypothesis in Reynolds-averaged Navier–Stokes models. Spanwise curvature was observed to decrease turbulent kinetic energy. These results have important implications on the development of turbulence in general applications, such as the flow over a prolate spheroid.

Previous studies show that at the small scales of stably stratified turbulence, the scale-dependent buoyancy flux reverses sign, corresponding to a conversion of turbulent potential energy (TPE) back into turbulent kinetic energy (TKE). Moreover, the magnitude of the reverse flux becomes stronger with increasing Prandtl number $\textit{Pr}$. Using a filtering analysis we demonstrate analytically how this flux reversal is connected to the mechanism identified in Bragg & de Bruyn Kops (2024 J. Fluid Mech. vol. 991, A10) that is responsible for the surprising observation that the TKE dissipation rate increases while the TPE dissipation rate decreases with increasing $\textit{Pr}$ in stratified turbulence. The mechanism identified by Bragg & de Bruyn Kops, which is connected to the formation of ramp–cliff structures in the density field, is shown to give the scale-local contribution to the buoyancy flux. At the smallest scales this local contribution dominates and explains the flux reversal, while at larger scales a non-local contribution is important. Direct numerical simulations of three-dimensional statistically stationary, strongly stably stratified turbulence confirm the theoretical analysis, and indicate that, while on average the local contribution only dominates the buoyancy flux at the smallest scales, it remains strongly correlated with the buoyancy flux at all scales. The results show that ramp cliffs are not only connected to the reversal of the local buoyancy flux but also the non-local part. At the small scales (approximately below the Ozmidov scale), ramp structures contribute exclusively to reverse buoyancy flux events, whereas cliff structures contribute to both forward and reverse buoyancy flux events.

This work uses large-eddy simulations to study the transition of a tulip flame stabilised by bubble vortex breakdown (BVB) mode towards a V flame stabilised by a conical vortex breakdown (CVB). The transition is triggered when the equivalence ratio is increased, resulting in a rise in temperature within the central recirculation zone (CRZ). Simultaneously, the pressure inside the CRZ bubble increases, while the average pressure inside the combustion chamber remains constant. This increase in pressure causes the CRZ bubble to open radially and expand, changing the vortex breakdown mode from BVB to CVB, and the flame shape from a tulip shape to a V shape. A criterion for a limit pressure inside the CRZ was then devised based on the radial momentum equation and the balance between centrifugal force and radial pressure gradient, found to control the radial motion of the CRZ. This criterion helped us to understand the main events of the transition, showing that, once the pressure inside the CRZ exceeds the limit given by the criterion, the flow topology changes from a BVB mode to a CVB mode. This transition highlights the differences between a BVB mode and a CVB mode, showing for the first time that there are characteristic pressure and velocity profiles for each mode in their swirling jets and CRZ. Finally, a significant achievement of this work is the identification of a novel mechanism for the controlled transition of vortex breakdown mode, a combustion-driven transition of vortex breakdown mode.

The modal and non-modal stability of laminar flow in a rectangular microchannel is investigated by incorporating the effects of Coriolis forces due to rotation, cross-sectional aspect ratio and superhydrophobic wall slip. The full Navier–Stokes equations are linearised into modified Orr–Sommerfeld and Squire equations, which are then formulated as an eigenvalue problem using small disturbances of the Tollmien–Schlichting type. These equations are subsequently solved by the spectral collocation method. The transition to instability in rotating microchannel flows, influenced by aspect ratio and slip conditions, is analysed through eigenvalue spectra and neutral stability curves. For non-modal analysis, we express the solution in matrix exponential form and then, using the singular value decomposition method, calculate the maximum energy growth. The study reveals that the flow becomes unstable in the presence of rotation at a critical Reynolds number of $ Re_c \approx 40$ for a low aspect ratio and $ Re_c \approx 50.4$ for a high aspect ratio. We find that instability is more pronounced in spanwise-rotating flows at higher aspect ratios compared with those at lower aspect ratios. Rotation induces disturbances from both walls along the spanwise direction, forming secondary flow structures near the centreline. Furthermore, we examine the influence of anisotropic slip by separately considering streamwise and spanwise slip as limiting cases. The numerical results demonstrate that while streamwise slip has a stabilising effect on rotating flows at small scales, a sufficiently large spanwise slip length can trigger instability at Reynolds numbers lower than those observed in the no-slip case. Rotation has the potential to enhance non-modal transient energy growth, while streamwise slip can effectively suppress this instability. These findings suggest that the onset of instability and transient energy growth can be effectively regulated by adjusting the aspect ratio and spanwise slip of the channel walls.