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.

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.

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.

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.

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.

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.

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.

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.

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.

Secondary flows induced by spanwise heterogeneous surface roughness play a crucial role in determining engineering-relevant metrics such as surface drag, convective heat transfer and the transport of airborne scalars. While much of the existing literature has focused on idealized configurations with regularly spaced roughness elements, real-world surfaces often feature irregularities, clustering and topographic complexity for which the secondary flow response remains poorly understood. Motivated by this gap, we investigate multicolumn roughness configurations that serve as a regularized analogue of roughness clustering. Using large-eddy simulations, we systematically examine secondary flows across a controlled set of configurations in which cluster density and local arrangement are varied in an idealized manner, and observe that these variations give rise to distinct secondary flow polarities. Through a focused parameter study, we identify the spanwise gap between the edge-most roughness elements of adjacent columns, normalized by the channel half-height ($s_a/H$), as a key geometric factor governing this polarity. In addition to analysing the time-averaged structure, we investigate how variations in polarity affect the instantaneous dynamics of secondary flows. Here, we find that the regions of high- and low-momentum fluid created by the secondary flows alternate in a chaotic, non-periodic manner over time. Further analysis of the vertical velocity signal shows that variability in vertical momentum transport is a persistent and intrinsic feature of secondary flow dynamics. Taken together, these findings provide a comprehensive picture of how the geometric arrangement of roughness elements governs both the mean structure and temporal behaviour of secondary flows.

The emergence of large-scale spatial modulations of turbulent channel flow, as the Reynolds number is decreased, is addressed numerically using the framework of linear stability analysis. Such modulations are known as the precursors of laminar–turbulent patterns found near the onset of relaminarisation. A synthetic two-dimensional base flow is constructed by adding finite-amplitude streaks to the turbulent mean flow. The streak mode is chosen as the leading resolvent mode from linear response theory. In addition, turbulent fluctuations can be taken into account or not by using a simple Cess eddy viscosity model. The linear stability of the base flow is considered by searching for unstable eigenmodes with wavelengths larger than the base flow streaks. As the streak amplitude is increased in the presence of the turbulent closure, the base flow loses its stability to a large-scale modulation below a critical Reynolds-number value. The structure of the corresponding eigenmode, its critical Reynolds number, its critical angle and its wavelengths are all fully consistent with the onset of turbulent modulations from the literature. The existence of a threshold value of the Reynolds number is directly related to the presence of an eddy viscosity, and is justified using an energy budget. The values of the critical streak amplitudes are discussed in relation with those relevant to turbulent flows.

Sea surface films significantly influence air–sea interaction. While their damping effect on gravity–capillary waves is well recognised, the detailed mechanisms by which surface films alter small-scale wave dynamics – particularly energy dissipation and near-surface flow patterns – remain insufficiently understood. This paper presents experimental observations focusing on small-scale wave profiles and surface-flow dynamics in the presence of surfactants, providing direct experimental evidence of underlying mechanisms such as Marangoni effects. The experiments demonstrate enhanced energy dissipation and significant alterations in near-surface flow caused by surfactants, including the transformation of typical circular motion into elliptical-like trajectories and the emergence of reverse surface drift.

A thin, evaporating sessile droplet with a pinned contact line containing inert particles is considered. In the limit in which the liquid flow decouples from the particle transport, we discuss the interplay between particle advection, diffusion and adsorption onto the solid substrate on which the droplet sits. We perform an asymptotic analysis in the physically relevant regime in which the Péclet number is large, i.e. ${\textit{Pe}}\gg 1$, so that advection dominates diffusion in the droplet except in a boundary layer near the contact line, and in which the ratio of the particle velocities due to substrate adsorption and diffusion is at most of order unity as ${\textit{Pe}}\rightarrow \infty$. We use the asymptotic model alongside numerical simulations to demonstrate that substrate adsorption leads to a different leading-order distribution of particle mass compared with cases with negligible substrate adsorption, with a significant reduction of the mass in the suspension – the nascent coffee ring reported in Moore et al. (J. Fluid Mech., vol. 920, 2021, A54). The redistribution leads to an extension of the validity of the dilute suspension assumption, albeit at the cost of breakdown due to the growth of the deposited layer, which are important considerations for future models that seek to accurately model the porous deposit regions.

A fully resolved numerical study was performed to investigate interfacial heat and mass transfer enhanced by the fully developed Rayleigh–Bénard–Marangoni instability in a relatively deep domain. The instability was triggered by evaporative cooling modelled by a constant surface heat flux. The latter allowed for temperature-induced variations in surface tension giving rise to Marangoni forces reinforcing the Rayleigh instability. Simulations were performed at a fixed Rayleigh number (${\textit{Ra}}_h$) and a variety of Marangoni numbers (${\textit{Ma}}_h$). In each simulation, scalar transport equations for heat and mass concentration at various Schmidt numbers (${\textit{Sc}}=16{-}200$) were solved simultaneously. Due to the fixed (warm) temperature prescribed at the bottom of the computational domain, large buoyant plumes emerged quasi-periodically both at the top and bottom. With increasing Marangoni number a decrease in the average convection cell size at the surface was observed, with a simultaneous improvement in near-surface mixing. The presence of high aspect ratio rectangular convection cell footprints was found to be characteristic for Marangoni-dominated flows. Due to the promotion of interfacial mass transfer by Marangoni forces, the power in the scaling of the mass transfer velocity, $K_{\!L}\!\propto\! \textit{Sc}^{-n}$, was found to decrease from $n=0.50$ at ${\textit{Ma}}_h=0$ to $\approx 0.438$ at ${\textit{Ma}}_h=13.21\times 10^5$. Finally, the existence of a buoyancy-dominated and a Marangoni-dominated regime was investigated in the context of the interfacial heat and mass transfer scaling as a function of ${\textit{Ma}}_h+\varepsilon {\textit{Ra}}_h$, where $\varepsilon$ is a small number determined empirically.

As a step towards realising a skin-friction drag reduction technique that scales favourably with Reynolds number, the impact of a synthetic jet on a turbulent boundary layer was explored through a study combining wind-tunnel measurements and large eddy simulations. The jet was ejected in the wall-normal direction through a rectangular slot whose spanwise dimension matched that of dominant large-scale structures in the logarithmic region to target structures of that size and smaller simultaneously. Local skin-friction reduction was observed at both $x/\delta =2$ and $x/\delta =5$ downstream of the orifice centreline, where $\delta$ is the boundary-layer thickness. At $x/\delta =2$, the skin-friction reduction was observed to be due to the synthetic-jet velocity deficit intersecting the wall. At $x/\delta =5$, evidence from the simulations and wind-tunnel measurements suggests that a weakening of wall-coherent velocity scales is primarily responsible for the skin-friction reduction. Local skin-friction reduction which scales favourably with Reynolds number may be achievable with the synthetic jet employed in this study. However, there are many technical hurdles to overcome to achieve net skin-friction drag reduction over the entire region of influence. For instance, regions of skin-friction increase were observed close to the orifice ($x/\delta \lt 2$) and downstream of the orifice edge due to the induced motion of synthetic-jet vortical structures. Additionally, a recirculation region was seen to form during expulsion, which has implications for pressure drag on non-planar surfaces.

Both experiments and direct numerical simulation (DNS) of hypersonic flow over a compression ramp show streamwise aligned streaks/vortices near the corner as the ramp angle is increased. The origin of this three-dimensional disturbance growth is not definitively known in the existing literature, but is typically connected to flow deceleration, centrifugal (Görtler) and/or baroclinic effects. In this work we consider the hypersonic problem with moderate wall cooling in the high Reynolds/Mach number, weak interaction limit. In the lower deck of the corresponding asymptotic triple-deck description we pose the linearised, three-dimensional, Görtler stability equations. This formulation allows computation of both receptivity and biglobal stability problems for linear spanwise-periodic disturbances with a spanwise wavelength of the same order as the lower-deck depth. In this framework the dominant response near the ramp surface is of constant density and temperature (at leading order) ruling out baroclinic mechanisms. Nevertheless, we show that there remains strong energy growth of upstream spanwise-varying perturbations and ultimately a bifurcation from two-dimensional to three-dimensional ramp flow. The unstable eigenmodes are localised to the separation region. The bifurcation points are obtained over a range of ramp angle, wall-cooling parameter and disturbance wavelength. Consistent with DNS results, the three-dimensional perturbations in this asymptotic formulation are streamwise aligned streaks/vortices, displaced above the separation region. In addition, the growth of upstream disturbances peaks near to the reattachment point, whilst the streaks persist beyond it, decaying relatively slowly downstream along the deflected ramp.

We propose a novel multiple-scale spatial marching method for flows with slow streamwise variation. The key idea is to couple the boundary region equations, which govern large-scale flow evolution, with local exact coherent structures that capture the small-scale dynamics. This framework is consistent with high-Reynolds-number asymptotic theory and offers a promising approach to constructing time-periodic finite-amplitude solutions in a broad class of spatially developing shear flows. As a first application, we consider a non-uniformly curved channel flow, assuming that a finite-amplitude travelling-wave solution of plane Poiseuille flow is sustained at the inlet. The method allows for the estimation of momentum transport and highlights the impact of the inlet condition on both the transport properties and the overall flow structure. We then consider a case with gradually decreasing curvature, starting with Dean vortices at the inlet. In this setting, small external oscillatory disturbances can give rise to subcritical self-sustained states that persist even after the curvature vanishes.

The flow past a $6:1$ prolate spheroid at a moderate pitch angle $\alpha =10^\circ$ is investigated with a focus on the turbulent wake in a high-fidelity large eddy simulation (LES) study. Two length-based Reynolds numbers, ${\textit{Re}}_L=3\times 10^4$ and $9\times 10^4$, and four Froude numbers, ${\textit{Fr}} = \infty \text{(unstratified)}, 6, 1.9 \text{ and }1$, are selected for the parametric study. Spectral proper orthogonal decomposition (SPOD) analysis of the flow reveals the leading coherent modes in the unsteady separated flow at the tail of the body. At the higher ${\textit{Re}}_L=9\times 10^4$, a high-frequency spanwise flapping of shear layers on either side of the body is observed in the separated boundary layer for all cases. The flapping does not perturb the lateral symmetry of the wake. At ${\textit{Fr}}=\infty$, a low-frequency oscillating laterally asymmetric mode, which is found in addition to the shear-layer mode, leads to a sidewise unsteady lateral load. All temporally averaged wakes at ${\textit{Re}}=9\times 10^4$ are found to be spanwise symmetric in the mean as opposed to the lower ${\textit{Re}}=3\times 10^4$, at which the ${\textit{Fr}}=\infty \text{ and }6$ wakes exhibit asymmetry. The turbulent kinetic energy (TKE) budget is compared among cases. Here, ${\textit{Fr}}=\infty$ exhibits higher production and dissipation compared with ${\textit{Fr}}=6 \text{ and }1.9$. The streamwise vortex pair in the wake induces a significant mean vertical velocity ($U_z$). Therefore, in contrast to straight-on flow, the terms involving gradients of $U_z$ matter to TKE production. Buoyancy reduces $U_z$ and also the Reynolds shear stresses involving $u^{\prime}_z$. Through this indirect mechanism, buoyancy exerts control on the wake TKE budget, albeit being small relative to production and dissipation. Buoyancy, through the baroclinic torque, is found to qualitatively affect the streamwise vorticity. In particular, the primary vortex pair is extinguished in the intermediate wake and two new vortex pairs form with opposite-sense circulation relative to the primary.