The objective of this work is to investigate the unexplored laminar-to-turbulent transition of a heated flat-plate boundary layer with a fluid at supercritical pressure. Two temperature ranges are considered: a subcritical case, where the fluid remains entirely in the liquid-like regime, and a transcritical case, where the pseudo-critical (Widom) line is crossed and pseudo-boiling occurs. Fully compressible direct numerical simulations are used to study (i) the linear and nonlinear instabilities, (ii) the breakdown to turbulence, and (iii) the fully developed turbulent boundary layer. In the transcritical regime, two-dimensional forcing generates not only a train of billow-like structures around the Widom line, resembling Kelvin–Helmholtz instability, but also near-wall travelling regions of flow reversal. These spanwise-oriented billows dominate the early nonlinear stage. When high-amplitude subharmonic three-dimensional forcing is applied, staggered $\Lambda$-vortices emerge more abruptly than in the subcritical case. However, unlike the classic H-type breakdown under zero pressure gradient observed in ideal-gas and subcritical regimes, the H-type breakdown is triggered by strong shear layers caused by flow reversals – similar to that observed in adverse pressure gradient boundary layers. Without oblique wave forcing, transition is only slightly delayed and follows a naturally selected fundamental breakdown (K-type) scenario. Hence in the transcritical regime, it is possible to trigger nonlinearities and achieve transition to turbulence relatively early using only a single two-dimensional wave that strongly amplifies background noise. In the fully turbulent region, we demonstrate that variable-property scaling accurately predicts turbulent skin-friction and heat-transfer coefficients.

The present work aims at exploring the scale-by-scale kinetic energy exchanges in multiphase turbulence. For this purpose, we derive the Kármán–Howarth–Monin equation which accounts for the variations of density and viscosity across the two phases together with the effect of surface tension. We consider both conventional and phase conditional averaging operators. This framework is applied to numerical data from detailed simulations of forced homogeneous and isotropic turbulence covering different values for the liquid volume fraction, the liquid–gas density ratio, the Reynolds number and the Weber number. We confirm the existence of an additional transfer term due to surface tension. Part of the kinetic energy injected at large scales is transferred into kinetic energy at smaller scales by classical nonlinear transport while another part is transferred to surface energy before being released back into kinetic energy, but at smaller scales. The overall kinetic energy transfer rate is larger than in single-phase flows. Kinetic energy budgets conditioned in a given phase show that the scale-by-scale transport of turbulent kinetic energy due to pressure is a gain (loss) of kinetic energy for the lighter (heavier) phase. Its contribution can be dominant when the gas volume fraction becomes small or when the density ratio increases. Building on previous work, we hypothesise the existence of a pivotal scale above which kinetic energy is stored into surface deformation and below which the kinetic energy is released by interface restoration. Some phenomenological predictions for this scale are discussed.

The linear Faraday instability of a viscous liquid film on a vibrating substrate is analysed. The importance is in the first step in applications for ultrasonic liquid-film destabilisation. The equations of motion are linearised and solved for a liquid film with constant thickness vibrating in a direction normal to its interface with an ambient gaseous medium treated as dynamically inert. Motivated by empirical evidence and the weakly nonlinear analysis of Miles (J. Fluid Mech., vol. 248, 1993, pp. 671–683), we choose an ansatz that the free liquid-film surface forms a square-wave pattern with the same wavenumbers in the two horizontal directions. The result of the stability analysis is a complex rate factor in the time dependency of the film surface deformation caused by the vibrations at a given excitation frequency and vibration amplitude. The analysis allows Hopf bifurcations in the liquid-film behaviour to be identified. Regimes of the deformation wavenumber and the vibration amplitude characterised by unstable film behaviour are found. Inside the regimes, states with given values of the deformation growth rate are identified. The influence of all the governing parameters, such as the vibration amplitude and frequency, the deformation wavenumber and the liquid material properties, on the liquid-film stability is quantified. Non-dimensional relations for vibration amplitudes characteristic for changing stability behaviour are presented.

The turbulent evolution of the shallow water system exhibits asymmetry in vorticity. This emergent phenomenon can be classified as ‘balanced’, that is, it is not due to the inertial-gravity-wave modes. The quasi-geostrophic (QG) system, the canonical model for balanced motion, has a symmetric evolution of vorticity, thus misses this phenomenon. Here, we present a next-order-in-Rossby extension of QG, $\textrm {QG}^{+1}$, in the shallow water context. We recapitulate the derivation of the model in one-layer shallow water grounded in physical principles and provide a new formulation using ‘potentials’. Then, the multi-layer extension of the shallow water quasi-geostrophic equation ($\textrm {SWQG}^{+1}$) model is formulated for the first time. The $\textrm {SWQG}^{+1}$ system is still balanced in the sense that there is only one prognostic variable, potential vorticity (PV), and all other variables are diagnosed from PV. It filters out inertial-gravity waves by design. This feature is attractive for modelling the dynamics of balanced motions that dominate transport in geophysical systems. The diagnostic relations connect ageostrophic physical variables and extend the massively useful geostrophic balance. Simulations of these systems in classical set-ups provide evidence that $\textrm {SWQG}^{+1}$ captures the vorticity asymmetry in the shallow water system. Simulations of freely decaying turbulence in one layer show that $\textrm {SWQG}^{+1}$ can capture the negatively skewed vorticity, and simulations of the nonlinear evolution of a baroclinically unstable jet show that it can capture vorticity asymmetry and finite divergence of strain-driven fronts.

The interaction between cavitation bubbles and particles near rigid boundaries plays a crucial role in applications from surface cleaning to cavitation erosion. We present a combined experimental, numerical and theoretical investigation of how boundary layer flows affect particle motion during the growth and collapse of the cavitation bubble. Using laser-induced cavitation bubbles and particles of varying radius ratios and stand-off distances, we observe that increasing the bubble-to-particle size ratio suppresses particle displacement. Through one-way coupled simulations and theoretical modelling, we demonstrate that this suppression arises from a shift in the dominant forces acting on the particle: for small radius ratios, the pressure gradient force governs particle motion, while for large ratios, the interplay between added mass, lubrication, and pressure gradient forces becomes significant due to boundary layer growth in the bubble-induced stagnation flow. Based on a theoretical framework combining potential flow theory and axisymmetric viscous stagnation flow analysis, we identify the inviscid- and viscous-flow dominated regimes characterised by the combination of the stand-off distance, the bubble-to-particle radius ratio, and the bubble Reynolds number. Finally, we derive scaling laws for particle displacement consistent with experiments and simulations. These findings advance our understanding of unsteady boundary layer effects in cavitation bubble-particle interactions, offering new insights for applications in microparticle manipulation and flow measurements.

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.