For a perturbed trefoil vortex knot evolving under the Navier–Stokes equations, a sequence of $\nu$-independent times $t_m$ are identified that correspond to a set of scaled, volume-integrated vorticity moments $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}$, with this hierarchy $t_\infty \leqslant \ldots \leqslant t_m\ldots t_1=t_x\approx 40$ and $\mathcal{O}_{\textit{Vm}}=(\int _{V\ell }|\omega |^{2m}\,{\rm d}V)^{1/2m}$. For the volume-integrated enstrophy $Z(t)$, convergence of $\sqrt {\nu }Z(t)=\bigl (\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}}(t)\bigr )^2$ at $t_x=t_1$ marks the end of reconnection scaling. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. Meanwhile $Z$ accelerates, leading to approximate finite-time $\nu$-independent convergence of the energy dissipation rate $\epsilon (t)=\nu Z(t)$ at $t_\epsilon \sim 2t_x$. This is sustained over a finite temporal span of at least $\Delta T_\epsilon \searrow 0.5 t_\epsilon$, giving Reynolds number independent finite-time, temporally integrated dissipation, $\Delta E_\epsilon =\int _{\Delta T_\epsilon }\epsilon \,{\rm d}t$, and thus satisfies one definition for a dissipation anomaly, with enstrophy spectra that are consistent with transient $k^{1/3}$ Lundgren-like inertial scaling over some of the $\Delta T_\epsilon$ time. A critical factor in achieving these temporal convergences is how the computational domain $V_\ell =(2\ell \pi )^3$ is increased as $\ell \sim \nu ^{-1/4}$, for $\ell =2$ to 6, then to $\ell =12$, as $\nu$ decreases. Appendix A shows compatibility with established $(2\pi )^3$ mathematics where $\nu \equiv 0$ Euler solutions bound small $\nu$ Navier–Stokes solutions. Two spans of $\nu$ are considered. Over the first factor of 25 decrease in $\nu$, most of the $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease (125 total) in $\nu$, with increased $\ell$ to $\ell =12$, there is initially only convergence of $\nu ^{1/4}\varOmega _{V\infty }(t)$ to $t_\infty$, without convergence for $9\gt m\gt 1$. Nonetheless, there is later $\sqrt {\nu }Z(t)$ convergence at $t_1=t_x$ and $\epsilon (t)=\nu Z$ over $t\sim t_\epsilon \approx 2t_x$.

Bayesian optimisation with Gaussian process regression was performed to optimise the shape of an elastically mounted cylinder undergoing transverse flow-induced vibration. The vibration amplitude and mean power coefficient were obtained from two-dimensional numerical simulations, with Reynolds number $Re = 100$. First, shape optimisation was performed to maximise the amplitude of undamped vibrations. The optimised shape was found to be a thin crescent cylinder aligned perpendicular to the oncoming flow. The optimised shapes exhibited simultaneous vortex-induced vibration and galloping, a response which was not observed for other cylinder geometries at the same Reynolds number. Shape optimisation was also performed to maximise the power coefficient, where the power generation device was modelled as a linear damper. The power-optimised cylinders were also thin crescents, but with greater curvature compared with the amplitude-optimised cylinders. Compared with a circular cylinder, improvements in the power coefficient and efficiency of up to $523\,\%$ and $152\,\%$, respectively, were obtained.

The stability and dynamics of flows past axisymmetric bubble-shaped rigid bluff bodies have been numerically and experimentally investigated. Motivated by the shapes of bubbles rising in quiescent liquids the bluff bodies were modelled as spherical and elliptical caps. The geometries are characterised by their aspect ratio, $\chi$, defined as the ratio of the height of the bubble to the base radius, which is varied from $0.2$ to $2.0$. Linear stability analyses were carried out on axisymmetric base flow fields subject to three-dimensional perturbations. As observed in earlier studies on bluff-body wakes, the primary bifurcation is stationary, followed by an oscillatory secondary bifurcation, with the leading global mode corresponding to azimuthal wavenumber $m = 1$. The domain of stability is found to increase with aspect ratio for both of the geometries considered in the present study. The critical Reynolds number corresponding to the primary bifurcation is found to be independent of the aspect ratio when re-scaled using the extent of the recirculation region and the maximum of the reverse-flow velocity as the length and velocity scales, respectively. The wake flow features were characterised experimentally using laser-induced fluorescence and particle-image-velocimetry techniques. It is observed that the flow has a planar symmetry following the primary bifurcation, which is retained beyond the secondary bifurcation. The experimentally measured wavelengths and frequencies are in excellent agreement with the results obtained from global stability analyses. These observations were further corroborated using direct numerical simulations of the three-dimensional flow field. The critical Reynolds numbers corresponding to both primary and secondary bifurcations, and the dominant modes obtained using proper orthogonal decomposition of the experimentally measured velocity fields, are found to agree well with the global mode shapes and numerically computed flow fields.

We investigate how the addition of surfactant affects the governing equations for a bubble in a two-dimensional channel in the small-capillary-number limit. In the limit where the time scale for absorption of surfactant is much shorter than the time scales for transport of surfactant along the surface of the bubble, we derive a set of idealised free-surface boundary conditions that capture the effects of surfactant in a single dimensionless ‘elasticity parameter’, and apply them to the front and rear of the bubble separately. At the front of the bubble, there are three regions of interest: the front cap, the thin film region and a transition region that smoothly connects the other two regions. Through matched asymptotic expansions, we derive predictions for the thin film height and the pressure drop across the front meniscus. We find that the viscous pressure drop across the front meniscus is always larger for a surfactant-laden bubble than for a surfactant-free bubble, by an order-one factor of up to $4^{2/3}$. Using a similar analysis at the rear of the bubble, we find that the viscous pressure drop across the rear meniscus is also always larger in magnitude for a surfactant-laden bubble than for a surfactant-free bubble, again up to a maximum factor of $4^{2/3}$. Finally, we use these results to show that, for the same flow conditions, an isolated surfactant-laden bubble in a Hele-Shaw cell will travel more slowly than an isolated surfactant-free bubble.

Surface patterns on ablating materials are known to appear in both high-speed ground and flight tests, but the mechanisms behind their formation are not known. In this paper, the origins of surface patterns are investigated via a local linear stability analysis of compressible laminar boundary layers over a flat camphor plate. The effects of sublimation and conjugate heat transfer are included on both the baseflow and the linear fluctuations. This newly developed framework identifies a single mode that fully characterises the stability of the surface, and this surface mode becomes unstable under laminar conditions only when the wall temperature exceeds that of an adiabatic wall, $T_{\textit{ad}}$. These findings are consistent with experimental observations, where laminar flow conditions at adiabatic wall temperatures are observed to be stable. The present analysis also reveals that the nature of this surface mode varies as a function of the oblique angle $\psi = \tan ^{-1}({\beta /\alpha })$, where $\alpha$ and $\beta$ are the streamwise and spanwise wavenumbers. As the wall temperature increases, the most unstable orientation of the surface mode shifts from streamwise alignment ($\psi = 0$), towards the sonic angle ($\psi = \psi _s = \cos ^{-1}(1/M_e)$) and then towards spanwise alignment ($\psi = 90^\circ$). Finally, a critical wavenumber is identified (i.e. one at which the temporal growth rate reaches a maximum) which implies the formation of a surface pattern of a specific wavelength and orientation.

Transient thermocapillary convection flows near a suddenly heated vertical wire are widely present in nature and industrial systems. The current study investigates the dynamical evolution and heat transfer for these transient flows near a suddenly heated vertical wire, employing scaling analysis and axisymmetric numerical simulation methodologies. Scaling analysis indicates that there exist four possible scenarios of the dynamical evolution and heat transfer for these transient flows, dependent on the wire curvature, Marangoni number and Prandtl number. In a typical scenario of the dynamical evolution and heat transfer, heat is first conducted into the fluid after sudden heating, resulting in an annular vertical thermal boundary layer around the wire. The radial temperature gradient may generate a thermocapillary force on the liquid surface, dragging the liquid away from the wire. The pressure gradient also drives a vertical flow along the wire. Further, the current study analyses and derives the scaling laws of the velocity, thickness and Nusselt number for the surface and vertical flows in different scenarios. Additionally, a number of two-dimensional axisymmetric numerical simulations are performed. The flow structure around the suddenly heated vertical wire is characterised under different regimes and the validation for the proposed scaling laws in comparison with numerical results is presented.

The dispersion of solutes has been extensively studied due to its important applications in microfluidic devices for mixing, separation and other related processes. Solute dispersion in fluids can be analysed over multiple time scales; however, Taylor dispersion specifically addresses long-term behaviour, which is primarily influenced by advective dispersion. This study investigates Taylor–Aris dispersion in a viscoelastic fluid flowing through axisymmetric channels of arbitrary shape. The fluid’s rheology is described using the simplified Phan-Thien–Tanner (sPTT) model. Although the channel walls are axisymmetric, they can adopt any geometry, provided they maintain small axial slopes. Drawing inspiration from the work of Chang & Santiago (2023 J. Fluid Mech. vol. 976, p. A30) on Newtonian fluids, we have developed a governing equation for solute dynamics that accounts for the combined effects of fluid viscoelasticity, molecular diffusivity and channel geometry. This equation is expressed using key dimensionless parameters: the Weissenberg number, the Péclet number and a shape-dependent dimensionless function. Solving this model allows us to analyse the temporal evolution of the solute distribution, including its mean and variance. Our analysis shows that viscoelasticity significantly decreases the effective solute diffusivity compared with that observed in a Newtonian fluid. Additionally, we have identified a specific combination of parameters that results in zero or negative transient growth of the variance. This finding is illustrated in a phase diagram and provides a means for transient control over dispersion. We validated our results against Brownian dynamics simulations and previous literature, highlighting potential applications for the design and optimisation of microfluidic devices.

The spatial organisation of a passive scalar plume originating from a point source in a turbulent boundary layer is studied to understand its meandering characteristics. We focus shortly downstream of the isokinetic injection ($1.5\leqslant x/\delta \leqslant 3$, $\delta$ being the boundary-layer thickness) where the scalar concentration is highly intermittent, the plume rapidly meanders and breaks up into concentrated scalar pockets due to the action of turbulent structures. Two injection locations were considered: the centre of the logarithmic region and the wake region of the boundary layer. Simultaneous quantitative acetone planar laser-induced fluorescence and particle image velocimetry were performed in a wind tunnel, to measure scalar mixture fraction and velocity fields. Single- and multi-point statistics were compared with established works to validate the diagnostic novelties. Additionally, the spatial characteristics of plume intermittency were quantified using ‘blob’ size, shape, orientation and mean concentration. It was observed that straining, breakup and spatial reorganisation were the primary plume-evolution modes in this region, with little small-scale homogenisation. Further, the dominant role of coherent vortex motions in plume meandering and breakup was evident. Their action is found to be the primary mechanism by which the injected scalar is transported away from the wall in high concentrations (‘large meander events’). Strong spatial correlation was observed in both instantaneous and conditional fields between the high-concentration regions and individual vortex heads. This coherent transport was weaker for wake injection, where the plume only interacts with outer vortex motions. A coherent-structure-based mechanism is suggested to explain these transport mechanisms.

New experimental results on gas flow through a long tube in the viscous, slip and transitional regimes are presented, obtained using an improved constant-volume measurement technique. This method is based on measuring the pressure variation in the inlet tank while the outlet tank is evacuated to a low pressure. Experimental pressure data for helium, neon, argon, nitrogen, krypton and xenon are used to extract the Poiseuille coefficient through a newly developed methodology. The obtained values show good agreement with theoretical predictions. Additionally, the velocity slip coefficient is also extracted from the same pressure data for all tested gases.

Heat-transfer measurements published in the literature seem to be contradictory, some showing a transition for the dependance of the Nusselt number (${\textit{Nu}}$) with the Rayleigh number (${\textit{Ra}}$) behaviour at ${\textit{Ra}} \approx 10^{11}$, some showing a delayed transition at higher ${\textit{Ra}}$, or no transition at all. The physical origin of this discrepancy remains elusive, but is hypothesised to be a signature of the multiple possible flow configurations for a given set of control parameters, as well as the sub-critical nature of the transition to the ultimate regime (Roche 2020 New J. Phys. vol. 22, 073056; Lohse & Shishkina 2023 Phys. Today vol. 76, no. 11, 26–32). New experimental and numerical heat-flux and velocity measurements, both reaching ${\textit{Ra}}$ up to $10^{12}$, are reported for a wide range of operating conditions, with either smooth boundaries, or mixed smooth–rough boundaries. Experiments are run in water at $40\,^\circ \textrm {C}$ (Prandtl number, ${\textit{Pr}}$ is 4.4), or fluorocarbon at $40\,^\circ \textrm {C}$ (${\textit{Pr}}$ is 12), and aspect ratios 1 or 2. Numerical simulations implement the Boussinesq equations in a closed rectangular cavity with a Prandtl number 4.4, close to the experimental set-up, also with smooth boundaries, or mixed smooth–rough boundaries. In the new measurements in the rough part of the cell, the Nusselt number is compatible with a ${\textit{Ra}}^{1/2}$ scaling (with logarithmic corrections), hinting at a purely inertial regime. In contrast to the ${\textit{Nu}}$ vs ${\textit{Ra}}$ relationship, we evidence that these seemingly different regimes can be reconciled: the heat flux, expressed as the flux Rayleigh number, ${\textit{Ra}}\textit{Nu}$, recovers a universal scaling with Reynolds number, which collapses all data, both our own and those in the literature, once a universal critical Reynolds number is exceeded. This universal collapse can be related to the universal dissipation anomaly, observed in many turbulent flows (Dubrulle 2019 J. Fluid Mech. vol. 867, no. P1, 1).

Whilst surface-stress integration remains the standard approach for fluid force evaluation, control-volume integral methods provide deeper physical insights through functional relationships between the flow field and the resultant force. In this work, by introducing a second-order tensor weight function into the Navier–Stokes equations, we develop a novel weighted-integral framework that offers greater flexibility and enhanced capability for fluid force diagnostics in incompressible flows. Firstly, in addition to the total force and moment, the weighted integral methods establish, for the first time, rigorous quantitative connections between the surface-stress distribution and the flow field, providing potential advantages for flexible body analyses. Secondly, the weighted integral methods offer alternative perspectives on force mechanisms, through vorticity dynamics or pressure view, when the weight function is set as divergence-free or curl-free, respectively. Thirdly, the derivative moment transformation (DMT)-based integral methods (Wu et al., J. Fluid Mech. vol. 576, 2007, 265–286) are generalised to weighted formulations, by which the interconnections among the three DMT methods are clarified. In the canonical problem of uniform flow past a circular cylinder, weighted integral methods demonstrate advantages in yielding new force expressions, improving numerical accuracy over original DMT methods, and enhancing surface-stress analysis. Finally, a force expression is derived that relies solely on velocity and acceleration at discrete points, without spatial derivatives, offering significant value for experimental force estimation. This weighted integral framework holds significant promise for flow diagnostics in fundamentals and applications.

The merging of two turbulent fronts without mean shear is investigated by direct numerical simulations. The turbulent streams are created by prescribing instantaneous velocity fields from precursor simulations of homogeneous isotropic turbulence (HIT) as inlet conditions for spatially evolving turbulent merging. The fronts are initially separated by a distance $H$ and convected with a uniform free stream velocity $U_{\infty }$. The inlet turbulence intensity varies in the range of $0.24 \leqslant u^{\prime}/U_{\infty } \leqslant 0.47$, while the inlet Taylor-scale Reynolds number is in the range of $151 \leqslant \textit{Re}_{\lambda } \leqslant 317$. As the flow develops in the streamwise direction, two distinct regions are identified: (i) an initial linear decay region, where the two turbulent fronts gradually approach each other without noticeable interaction; and (ii) a rapid decay region, where the opposing turbulent fronts influence one another and eventually merge. The flow statistics collapse once the streamwise coordinate is rescaled as $x^{+} = (x/H) (u^{\prime}/U_{\infty })$, suggesting that the merging location is imposed by large scales. An analysis conditioned to the developing turbulent/non-turbulent interfaces (TNTIs) reveals that, within the merging region, conditional mean enstrophy profiles deviate from those observed in ‘classical’ TNTIs, indicating a locally more homogenous flow. Within this region of interaction, the surface area of the TNTI increases while the volume of irrotational fluid steadily decreases, resulting in the generation of fine-scale structures. These findings support that turbulent merging is a multiscale process, where both the largest and smallest scales of motion intervene.