By deriving the Euler equations and Rankine–Hugoniot equations in the orthogonal frame field of the shock surface, the three-dimensional curved shock theory based on orthogonal frame of shock surface (3D-CST-boos) is established. In steady flow, this theory can be applied to three-dimensional (3-D) shocks without constraints on the incoming flow conditions. The derived equations elucidate the relationship between the first-order gradients of the preshock and postshock flow parameters and the geometric properties (curvature) of the 3-D curved shock. The correctness of 3D-CST-boos is verified for two-dimensional plane shocks and axisymmetric shocks. The analysis is then extended to the flow patterns of 3-D elliptical convex/concave shocks. Variations in the flow field behind a 3-D elliptical convex shock are explained based on different incoming flow conditions. Simultaneously, the fundamental mechanics underlying the differences between the flow fields of elliptical concave shocks and axisymmetric concave shocks are revealed using 3D-CST-boos. Finally, a concise analysis of the first-order flow parameters is presented for more complex 3-D shocks, including saddle-shaped shocks and cubic surface shocks.

Doubly diffusive convection describes the fluid motion driven by the competing buoyancy forces generated by temperature and salinity gradients. While the resulting convective motions usually occupy the entire domain, parameter regions exist where the convection is spatially localised. Although well studied in planar geometries, spatially localised doubly diffusive convection has never been investigated in a spherical shell, a geometry of relevance to astrophysics. In this paper, numerical simulation is used to compute spatially localised solutions of doubly diffusive convection in an axisymmetric spherical shell. Several families of spatially localised solutions, named using variants of the word convecton, are found and their bifurcation diagram computed. The various convectons are distinguished by their symmetry and by whether they are localised at the poles or at the equator. We find that, because the convection rolls that develop in the spherical shell are not straight but curve around the inner sphere, their strength varies with latitude, making the system prone to spatial modulation. As a consequence, spatially periodic states do not form from primary bifurcations and localised states are forced to arise via imperfect bifurcations. While the direct relevance of this work is to doubly diffusive convection, parallels drawn with the Swift–Hohenberg equation suggest a wide applicability to other pattern-forming systems in similar geometries.

Accurately modelling wind turbine wakes is essential for optimising wind farm performance but remains a persistent challenge. While the dynamic wake meandering (DWM) model captures unsteady wake behaviour, it suffers from near-wake inaccuracies due to empirical closures. We propose a symbolic regression-enhanced DWM (SRDWM) framework that achieves equation-level closure by embedding symbolic expressions for volumetric forcing and boundary terms explicitly into governing equations. These physically consistent expressions are discovered from large-eddy simulations (LES) data using symbolic regression guided by a hierarchical, domain-informed decomposition strategy. A revised wake-added turbulence formulation is further introduced to enhance turbulence intensity predictions. Extensive verification across varying inflows shows that SRDWM accurately reproduces both mean wake characteristics and turbulent dynamics, achieving full spatiotemporal resolution with over three orders of magnitude speed-up compared to LES. The results highlight symbolic regression as a bridge between data and physics, enabling interpretable and generalisable modelling.

Results are presented of an experimental investigation into the levitation of spheres on thin layers of viscous fluid. In one set of experiments the layer is formed on a planar vertical wall and in a second investigation the sphere sits on a fluid layer on the inside of a rotating horizontal cylinder. The motion takes place at a set of fixed locations in the latter case whereas the sphere generally translates up or down the plane wall of the belt. Lubrication layers formed between the surfaces of the spheres and the walls induce slip. Two distinct states are identified, and excellent accord is found between experimental results and those from a recently developed theory for the single-track state which is only observed in the rotating horizontal cylinder. The two-track state exists in both sets of experiments, but theoretical progress with this remains an outstanding challenge.

Transition onset of high-speed boundary layers can move first downstream and then upstream with increasing nose-tip bluntness, which is called transition reversal. For the first time, our recent research reproduced the experimentally observed transition reversal by direct numerical simulation (DNS, Guo et al., J. Fluid Mech. vol. 1005, 2025, A5). As a continuation study, this work explores the effect of the form of free-stream disturbances, as the transition in the large-bluntness regime still remains poorly understood. The free-stream Mach number is 5 and the nose-tip radius 3 mm of the blunt plate exceeds the experimental reversal value. Three-dimensional broadband perturbation is carefully constructed through superimposition of planar fundamental waves in the free stream, which initiates the transition in DNS. For each Fourier component, the same perturbation strength is applied for slow/fast acoustic, vortical and entropic waves. All the cases present a ‘streak-turbulent spot’ two-stage transition scenario due to non-modal instabilities. The transition onset locations induced by entropic and slow/fast acoustic waves are close and significantly ahead of that by vortical waves. More evident impact of the disturbance form is manifested in the length of the transitional region, which is the shortest for entropic waves and the longest for vortical waves. Regarding the effect of the angle of incidence that mimics the tunnel environment, it alters the post-shock acoustic-wave structure and reduces the length of the transitional region. In the streaky stage, the form of free-stream disturbances changes the pronounced spanwise wavelengths on the blunt nose and the plate, where the two regions also differ from each other. In the turbulent-spot region, the shortest transitional region induced by the entropic wave is attributed to its largest mean spanwise spreading rate of the turbulent spot. From the perspective of energy budget, shear-induced dissipation dominates the heat transfer escalation in the transitional region. Overall, with significant leading-edge bluntness, the flight environment may tend to result in delayed transition onset compared with the tunnel counterpart.

Electrical effects are known to play an important role in particle-laden flows, yet a holistic view of how they modulate turbulence remains elusive due to the complexity of multifield coupling. Here, we present a total of 119 direct numerical simulations of particle-laden turbulent channel flow that reveal a striking ability of electrical effects to induce turbulence relaminarisation and markedly alter wall drag. As expected, the transition from turbulence to laminar flow is accompanied by abrupt changes in the statistical properties of both the fluid and particulate phases. Nevertheless, with increasing electrical effects, the wall-normal profiles of the mean streamwise fluid velocity and mean local particle mass loading exhibit opposite trends in the turbulent and laminar regimes, arising from the competition between turbophoresis and electrostatic drift. We identify three distinct flow regimes resulting from the electrical effects: a drag-reduced turbulent regime, a drag-reduced laminar regime, and a drag-enhanced laminar regime. It is revealed that relaminarization originates from the complete suppression of the streak breakdown in the near-wall self-sustaining cycle, followed by the sequential inhibition of other subprocesses in the cycle. In the turbulent regime, increasing electrical effects induce opposing trends in Reynolds and particle stress contributions to drag, yielding a non-monotonic drag response. In laminar regimes, by contrast, the drag coefficient increases monotonically as the Reynolds stress vanishes and particle-induced stress becomes dominant.

We investigate the effect of inertial particles on Rayleigh-Bénard convection using weakly nonlinear stability analysis. An Euler–Euler/two-fluid formulation is used to describe the flow instabilities in particle-laden Rayleigh–Bénard convection. The weakly nonlinear results are presented near the critical point (bifurcation point) for water droplets in the dry air system. We show that supercritical bifurcation is the only type of bifurcation beyond the critical point in particle-laden Rayleigh–Bénard convection. Interaction of settling particles with the flow and the Reynolds stress or distortion terms emerges due to the nonlinear self-interaction of fundamental modes breaking down the top–bottom symmetry of the secondary flow structures. In addition to the distortion functions, the nonlinear interaction of fundamental modes generates higher harmonics, leading to the tendency of preferential concentration of uniformly distributed particles, which is completely absent in the linear stability analysis. Further, we show that in the presence of thermal energy coupling between the fluid and particles, the difference between the horizontally averaged heat flux at the hot and cold surfaces is equal to the net sensible heat flux advected by the particles. The difference between the heat fluxes at hot and cold surfaces increases with an increase in particle concentration.

Three-dimensional laminar flow over an inclined spinning disk is investigated at a Reynolds number of ${\textit{Re}} = 500$ and an angle of attack of $\alpha = 25^\circ$, for tip-speed ratios up to 3. Numerical simulations are performed to investigate the effect of spin on the aerodynamics and characterise the instabilities that occur. Increasing tip-speed ratio significantly increases both lift and drag monotonically. Several distinct wake regimes are observed, including vortex shedding in the non-spinning case, vortex-shedding suppression at moderate tip-speed ratios and a distinct corkscrew-like short-wavelength instability in the advancing tip vortex at higher tip-speed ratios. Vorticity generated by the spinning disk strengthens the advancing tip vortex, inducing a spanwise stretching in the trailing-edge vortex sheet. This helps to dissipate the vorticity, which in turn prevents roll up and suppresses vortex shedding. The short-wavelength instability shows qualitative and quantitative matches to the $(-2,0,1)$ principal mode of the elliptic instabilities seen in pairs of counter-rotating Batchelor vortices. The addition of vorticity from the disk rotation significantly alters the circulation and axial velocity in the tip vortices, giving rise to elliptic instability despite its absence in the non-spinning case. In select cases, lock-in between the frequency of the elliptic instability and twice the spin frequency is observed, indicating that disk rotation acts as an additional forcing for the elliptic instability. Additional simulations at different Reynolds numbers and angle of attacks are considered to examine the robustness of observed phenomena across different parameter combinations.

The Monin–Obukhov similarity theory (MOST) is a cornerstone of atmospheric science for describing turbulence in stable boundary layers. Extending MOST to stably stratified turbulent channel flows, however, is non-trivial due to confinement by solid walls. In this study, we investigate the applicability of MOST in closed channels and identify where and to what extent the theory remains valid. A key finding is that the ratio of the half-channel height to the Obukhov length serves as a governing parameter for identifying distinct flow regions and determining their corresponding mean velocity scaling. Hence, we propose a relation to estimate this ratio directly from the governing input parameters: the friction Reynolds and friction Richardson numbers ($\textit{Re}_{\tau }$ and $Ri_{\tau }$). The framework is tested against a series of direct numerical simulations across a range of $\textit{Re}_{\tau }$ and $Ri_{\tau }$. The reconstructed velocity profiles enable accurate prediction of the skin-friction coefficient crucial for quantifying pressure losses in stratified flows in engineering applications.

In this paper, we numerically investigate the orbit dynamics of three-dimensional symmetric Janus drops in shear flow using an improved ternary-fluids phase field method, focusing on how drop deformation and initial orientation affect the orbit drift of two configurations of Janus drops: dumbbell-shaped and near-spherical. We find that the motion of dumbbell-shaped drops eventually evolves into tumbling, while near-spherical drops attain stable spinning. We attribute this bifurcation in orbit drift to contrasting deformation dynamics and shape-dependent hydrodynamics of the two configurations. Specifically, the drift bifurcation is closely related to the aspect ratio of Janus drops at equilibrium, giving rise to two distinct mechanisms: (1) coupling between outer interface deformation and the surrounding flow field; and (2) interplay between inner interface deformation and vortices enclosed within the drop. In addition, we observe that for the dumbbell-shaped Janus drops with different aspect ratios, their tumbling dynamics resembles ellipsoids in shear flow. Moreover, the trajectories of the dumbbell-shaped Janus drops during orbit drift collapse onto a universal curve, independent of their initial orientations, and significant deformation and inertia accelerate the orbit transition. To quantitatively evaluate the effect of drop deformation on the orbit drift of the dumbbell-shaped Janus drops, we propose an effective aspect ratio model based on the drop shapes at equilibrium and at the maximum elongation. By incorporating the effective aspect ratio into Jeffery’s theory for solid particles, we accurately predict the rotation period and angular velocity of Janus drops in the tumbling regime and during the orbit drift, especially for drops with linear deformation. Moreover, the orbit parameter $C$ is found to vary exponentially with time for drops with linear deformation, while the time variation of $C$ transits from one exponential function to another for drops with nonlinear deformation.

The thermal interactions of liquid droplets impacting a moving substrate are investigated, combining theoretical modelling with experimental validation. An analytical model is developed to predict the time-evolving contact temperature and heat flux at the droplet–substrate interface. Accounting for the convective heat transport induced by the impacting drop, the model incorporates a finite thermal contact resistance, which is a critical parameter that was often neglected in earlier studies for drop impact. High-speed, spatially resolved infrared thermography is used to record the two-dimensional, transient temperature evolution at the droplet–substrate interface during drop impact on a rotating disc. Measured temperature maps are used for numerical simulations to reconstruct local interfacial heat fluxes. The model is validated for different droplet diameters, substrate velocities and thermal conditions. The findings demonstrate that the substrate velocity and droplet diameter have negligible influence on the thermal behaviour within the tested parameter space.

The deposition of droplets onto a swollen polymer network induces the formation of a wetting ridge at the contact line. Current models typically consider either viscoelastic effects or poroelastic effects, while polymeric gels often exhibit both properties. In this study, we investigate the growth of the wetting ridge using a comprehensive large-deformation theory that integrates both dissipative mechanisms – viscoelasticity and poroelasticity. In the purely poroelastic case, following an initial instantaneous incompressible deformation, the growth dynamics exhibits scale-free behaviour, independent of the elastocapillary length or system size. A boundary layer of solvent imbibition between the solid surface (in contact with the reservoir) and the region of minimal chemical potential is created. At later times, the ridge equilibrates on the diffusion time scale given by the elastocapillary length. When viscoelastic properties are incorporated, our findings show that, during the early stages (prior to the viscoelastic relaxation time scale), viscoelastic effects dominate the growth dynamics of the ridge and solvent transport is significantly suppressed. Beyond the relaxation time, the late-time dynamics closely resembles that of the purely poroelastic case. These findings are discussed in light of recent experiments, showing how our approach offers a new interpretation framework for wetting of polymer networks of increasing complexity.

The two-dimensional (2-D) evolution of perturbed long weakly nonlinear surface plane, ring and hybrid waves, consisting, to leading order, of a part of a ring and two tangent plane waves, is modelled numerically within the scope of the 2-D Boussinesq–Peregrine system. Numerical runs are initiated and interpreted using the reduced 2-D cylindrical Korteweg–de Vries (cKdV)-type and Kadomtsev–Petviashvili II (KPII) equations. The cKdV-type equation leads to two different models, the KdV$\theta$, where $\theta$ stands for a polar angle, and cKdV equations, depending on whether we use the general or singular (i.e. the envelope of the general) solution of the associated nonlinear first-order differential equation. The KdV$\theta$ equation is also derived directly from the 2-D Boussinesq–Peregrine system and used to analytically describe the intermediate 2-D asymptotics of line solitons subject to sufficiently long transverse perturbations of finite strength, while the cKdV equation is used to initiate outward- and inward-propagating ring waves with localised and periodic perturbations. Both of these equations, together with the KPII equation, are used to model the evolution of hybrid waves, where we show, in particular, that large localised waves (lumps) can appear as transient (emerging and then disappearing) states in the evolution of inward-propagating waves, contributing to the possible mechanisms for the generation of rogue waves. Detailed comparisons are made between the key features of the non-stationary 2-D modelling and relevant predictions of the reduced equations.

In this work, we study the reaction-controlled dual bubbles ripening on a heterogeneous substrate with high surface wettability hysteresis, where the bubbles evolve with constant contact radius but varied contact angle. We first theoretically derived the governing kinetic equation of bubble curvature radius $R_B$, based on which we surprisingly found three possible ripening processes under six different conditions, i.e. the classical Ostwald ripening (the bubble with the larger curvature radius $R_B$ exhibits an increase in $R_B$, while the bubble with the smaller curvature radius $R_B$ experiences a decrease in $R_B$), the reversed ripening (converse to Ostwald ripening), and the consistent ripening ($R_B$ of both bubbles increases or reduces consistently). Further analyses from the aspects of chemical potential and free energy lead to an interesting finding that the $R_B$ of two bubbles finally reach egalitarianism, independently of different ripening processes. Numerical results obtained from two-phase lattice Boltzmann modelling demonstrate excellent agreement with theoretical predictions, specifically concerning the kinetic equation, the various ripening processes, and the egalitarianism of bubble radii $R_B$ after ripening completion.

This study investigates the dynamics of free-surface turbulence (FST) using direct numerical simulations (DNS). We focus on the energy exchange between the deformed free-surface and underlying turbulence, examining the influence of high Reynolds (${\textit{Re}}$) and Weber (${\textit{We}}$) numbers at low to moderate Froude (${\textit{Fr}}$) numbers. The two-fluid DNS of FST at the simulated conditions is able to incorporate air entrainment effects in a statistical steady state. Results reveal that a high ${\textit{We}}$ number primarily affects entrained bubble shapes (sphericity), while ${\textit{Fr}}$ significantly alters free-surface deformation, two-dimensional compressibility and turbulent kinetic energy (TKE) modulation. Vortical structures are mainly oriented parallel to the interface. At lower ${\textit{Fr}}$, kinetic energy is redistributed between horizontal and vertical components, aligning with rapid distortion theory, whereas higher ${\textit{Fr}}$ preserves isotropy near the surface. Evidence of a reverse or dual energy cascade is verified through third-order structure functions, with upscale transfer near the integral length scale, and enhanced vertical kinetic energy in upwelling eddies. Phase-based discrete wavelet transforms of TKE show weaker decay at the smallest scales near the interface, suggesting contributions from gravitational energy conversion and reduced dissipation. The wavelet energy spectra also exhibits different scaling laws across the wavenumber range, with a $-3$ slope within the inertial subrange. These findings highlight scale- and proximity-dependent effects on two-phase TKE transport, with implications for subgrid modelling.

In this experimental study, we investigate, for the first time, the structure and evolution of the near wake of a circular cylinder in a flowing soap film at the onset of vortex shedding. The study primarily focuses on the changes occurring within the recirculation bubble, along with the evolution of vortex shedding. A significantly large recirculation bubble forms behind the cylinder in the soap film environment, characterized by small-scale vortices along its edges, an observation that starkly contrasts with its three-dimensional counterparts. These small-scale vortices driven by the Kelvin–Helmholtz instability, further induce a transverse deflection of the recirculation bubble, leading to an intermittent generation of the wake vortices. The instantaneous velocity field in the wake is examined, highlighting the clear evidence of intermittency in vortex formation. The frequency and wavelength of the chain of small-scale vortices on the recirculation bubble is evaluated, and a functional relationship with the flow Reynolds number is determined. We believe this observation to be novel, potentially revealing a new pathway for understanding the two-dimensional transition in bluff-body wakes.

Slip effects on solid boundaries are common in complex fluids. Boundary depletion layers in polymer solutions can create apparent slip effects, which can in turn significantly impact the dynamics of moving bodies. Motivated by microswimmer locomotion in such environments, we derive a series of slip slender-body theories for filamentous bodies experiencing slip-like boundary conditions. Using Navier’s slip model, we derive three slip slender-body theories, linking the body’s velocity to the distribution of hydrodynamic forces. The models are shown to be consistent with each other and with existing numerical computations. As the slip length increases, we show that the drag parallel to the body decreases towards zero while the perpendicular drag remains finite, in a manner which we quantify. This reduction in drag ratio is shown to be inversely related to microswimmer mobility in two simple swimmer models. This increase could help rationalise empirically observed enhanced swimming in complex fluids.

This paper investigates the flow past a flexible splitter plate attached to the rear of a fixed circular cylinder at low Reynolds number 150. A systematic exploration of the plate length ($L/D$), flexibility coefficient ($S^{*}$) and mass ratio ($m^{*}$) reveals new laws and phenomena. The large-amplitude vibration of the structure is attributed to a resonance phenomenon induced by fluid–structure interaction. The modal decomposition indicates that resonance arises from the coupling between the first and second structural modes, where the excitation of the second structural mode plays a critical role. Due to the combined effects of added mass and periodic stiffness variations, the two modes become synchronised, oscillating at the same frequency while maintaining fixed phase difference $\pi /2$. This further results in the resonant frequency being locked at half of the second natural frequency, which is approximately three times the first natural frequency. A reduction in plate length and an increase in mass ratio are both associated with a narrower resonant locking range, while a higher mass ratio also shifts this range towards lower frequencies. A symmetry-breaking bifurcation is observed for cases with $L/D\leqslant 3.5$, whereas for $L/D=4.0$, the flow remains in a steady state with a stationary splitter plate prior to the onset of resonance. For cases with a short flexible plate and a high mass ratio, the shortened resonance interval causes the plate to return to the symmetry-breaking stage after resonance, gradually approaching an equilibrium position determined by the flow field characteristics at high flexibility coefficients.