We examine the dynamic interactions between the large-scale coherent motion and the small-scale turbulence in the passive scalar field of a circular cylinder wake, where the coherent motion exhibits strong periodicity. A combination of four X-wires and four cold wires was used to simultaneously measure the three velocity and temperature fluctuations at nominally the same location. Measurements were taken at $x/d=10$, 20 and 40 in the mean shear plane at Reynolds number 2500, based on the cylinder diameter $d$ and the free-stream velocity. The phase-averaging technique is used to distinguish the large-scale coherent motion from the stochastic motion, enabling the construction of phase-averaged structure functions of the passive scalar in the scale phase plane. The maximum of the coherent scalar $\tilde {\theta }$ closely aligns with the minima of the phase-averaged strain $\langle S \rangle$ and the vortex centre, suggesting that heat is contained within the interior of the vortex. The scale-by-scale distributions of the scalar variance and the streamwise velocity variance exhibit a similar phase dependence associated with the coherent motion. This dependence is perceptible even at the smallest scales. However, as the distance from the cylinder increases, the perceivable scale range decreases and eventually disappears. An expression is formulated to describe the time-averaged second-order structure function of coherent scalar and the time-averaged second-order mixed structure function between the coherent scalar and coherent streamwise velocity at $x/d= 10$ and 20, where the coherent motion is prominent. Furthermore, the scale-by-scale contribution of the coherent scalar variance to the total scalar variance is evaluated. Also, we derive the scale-by-scale scalar variance transport equations that account for the coherent motion in both general and isotropic formulations. Assuming local isotropy, it is found that the equation agrees approximately with the experimental data across all scales at $x/d= 40$. Finally, the differences between the scale-by-scale transport equation for the stochastic scalar variance and that for the stochastic turbulent kinetic energy are discussed.

Integral modelling of turbulent buoyant plumes is crucial for rapid predictions of plume characteristics. While the governing equations are typically derived using self-similarity and a Boussinesq approximation, these assumptions may not hold for plumes originating from finite-area sources with large density ratios. This work evaluates the accuracy of integral-scale models for non-Boussinesq lazy plumes using high-fidelity numerical simulations of turbulent helium plumes. We analyse the plume kinematics by computing vertical fluxes, plume radius and radial profiles, establishing some disparities between common practice and physical accuracy. We identify how the definition of the plume radius changes the perception of the plume structure when the flow is not self-similar and derive a relationship between the flux-based and threshold-based definitions without requiring self-similarity. We then examine the plume dynamics by evaluating the source terms from the governing plume equations. Our results support neglecting diffusive and viscous effects but emphasise the importance of the mean pressure gradient, even in the self-similar regime. Two coefficients need to be modelled: the well-known entrainment coefficient and the lesser-known momentum correction coefficient, which is a correction required for the momentum equation to account for self-similar and slender approximations. The momentum correction coefficient is found to be approximately constant and slightly greater than the assumed value of 1. The standard entrainment coefficient models perform well up to a local Richardson number three times the asymptotic value but overpredict entrainment for larger Richardson numbers. We propose a correction using the known finite limit of entrainment at infinite Richardson number.

The nonlinear Tollmien–Schlichting waves mechanism of subcritical transitional flow in quasi-two-dimensional flow and two-dimensional (2-D) plane Poiseuille flow have been investigated (Camobreco et al. 2023 J. Fluid Mech., vol. 963, p. R2; Huang et al. 2024 J. Fluid Mech., vol. 994, p. A6). However, the subcritical transitional flow threshold has remained unsolved for 2-D shear flows since the problem was proposed in Trefethen et al. (1993 Science vol. 261, no. 5121, pp. 578–584). In this study, we proposed a theoretical analysis based on the nonlinear non-modal analysis and asymptotic analysis to quantify the scaling law for subcritical transitional flow of 2-D plane Poiseuille flow. The subcritical transitional flow induced by the critical disturbance experiences the nonlinear edge state with invariant disturbance kinetic energy (Huang et al. 2024 J. Fluid Mech. vol. 994, p. A6). Consequently, the required magnitude along with the edge state is predicted by asymptotic analysis, and the a priori threshold is achieved theoretically. All stages are validated by the numerical minimal seeds of different channels. The proposed theory predicts that the scaling laws are $O(Re^{-11/3})$ and $O(\textit{Re}^{-7/3})$ for the critical disturbances and their edge state, respectively. While the numerical thresholds of the subcritical transitional flow are $ \textit{Re}^{-11/3 \pm 0.06}$ and $ \textit{Re}^{-7/3 \pm 0.05}$, respectively.

A burning droplet in normal gravity inevitably encounters buoyant convection set up by the flame, which can significantly impact its shrinkage kinetics traditionally described by the D2-law. However, the detailed mechanism governing droplet vapourisation under such self-generated flame-driven buoyant convection remains elusive. Here, we present both experimental and theoretical evidence highlighting the critical role of buoyant convection in droplet combustion. Experimentally, we precisely measure the values of the shrinkage exponent n for various liquid fuels, revealing a significant departure from the D2-law. While the measured n values consistently fall within the narrow range 2.6–2.7, they exhibit a slight increase with the fuel’s boiling point. A more general and in-depth theory is also developed to explain such small but systematic variations, revealing that differences in flow and thermal boundary layer structures – arising from varying combustion intensities – may account for the observed trends. Our theory predicts three distinct values of n, namely 2.6, 8/3 ≈ 2.67 and 35/13 ≈ 2.69, successfully capturing slight differences in n among various fuels. This is the first study demonstrating that the shrinkage kinetics in droplet vapourisation driven by flame-induced buoyant convection is nearly universal, determined solely by the underlying transport mechanisms, although these can be significantly altered due to their high susceptibility to detailed fuel chemistry and combustion kinetics. The present theoretical framework not only enables accurate prediction and control of burning droplet behaviour, but also is extendable to analyse more complex combustion processes involving a broader range of fuel types and flow conditions.

Particles in compressible shear flows experience lifting effects due to asymmetric pressure and viscous forces across the particle surface, rotation induced by asymmetric viscous forces (Magnus effect), and asymmetric compression and viscous effects if near a wall (wall effect). This work focuses on the lifting force on a solid spherical particle due to asymmetric pressure and shear stress distributions driven by density and velocity gradients. We show via direct numerical simulation and verify using scaling arguments that the lifting force in unbounded laminar compressible shear flows is a function of dynamic pressure gradient. We show that steady flow regimes demonstrate predictable lifting forces. Unsteady flow regimes demonstrate asymmetric vortex shedding which creates lift in directions not readily predictable. Thus, predicting lift requires the ability to predict wake structure. We develop approximate delineations between wake types at Reynolds numbers up to 20 000. We use the non-dimensional dynamic pressure gradient, Mach number, Reynolds number and predicted wake structure to develop a shear-induced lift model. The proposed model can be used in conjunction with a drag model to simulate particle motion in compressible shear flow.

The present study focuses on the influence of gas swirl on the spray behaviour from a two-fluid coaxial atomiser with high gas-to-liquid dynamic pressure ratios $M$ by varying both the liquid Reynolds number ${\textit{Re}}_l$ and the gas Weber number ${\textit{We}}_g$. The investigations identify the deviations of the carrier phase velocity fields, droplet distribution, and dispersion when swirl is introduced to the gas phase compared with the non-swirling conditions. The changes in the axial, radial and tangential velocities of the continuous phase due to the introduction of swirl are highlighted while retaining a self-similar behaviour. The slip velocity of the large droplets in swirling sprays is negative, unlike the known positive value for non-swirling sprays. The shape of the radial profiles of the mean drop size is investigated along ${\textit{We}}_g$, notably revealing an inflection point for swirling sprays at high-${\textit{We}}_g$ values. A global assessment of the drop size uncovered that swirl leads to its increase for low $M$ while assisting spray formation at high $M$. Additionally, the radial profiles of axial fluxes for swirling sprays have a wider bell-shaped curve compared with non-swirling sprays at high $M$, unlike the off-centre maxima found for low $M$. However, the mentioned dependencies of drop sizes and fluxes cannot be determined by $M$ solely for intermediate gas-to-liquid momentum ratios ($23\lt M\lt 46$), and vary with ${\textit{Re}}_l$ and ${\textit{We}}_g$. In addition, the response of at least the mean droplets at the edge of the spray to the large gas eddies shows a linear relation with swirl intensity.

Unsteady aerodynamic forces in flapping wings arise from complex, nonlinear flow structures that challenge predictive modelling. In this work, we introduce a data-driven framework that links experimentally observed flow structures to sectional pressure loads on physical grounds. The methodology combines proper orthogonal decomposition and quadratic stochastic estimation (QSE) to model and interpret these forces using phase-resolved velocity fields from particle image velocimetry measurements. The velocity data are decomposed in a wing-fixed frame to isolate dominant flow features, and pressure fields are reconstructed by solving the Poisson equation for incompressible flows. The relationship between velocity and pressure modes is captured through QSE, which accounts for nonlinear interactions and higher-order dynamics. We introduce an uncertainty-based convergence criterion to ensure model robustness. Applied to a flapping airfoil, the method predicts normal and axial forces with less than 6 % average error using only two velocity modes. The resulting model reveals an interpretable underlying mechanism: linear terms in the QSE model the circulatory force linked to the formation of vortices on the wing, while quadratic terms capture the nonlinear component due to added-mass effects and flow–vorticity interactions. This data-driven yet physically grounded approach offers a compact tool for modelling the unsteady aerodynamics in flapping systems with potential to generalise to other problems.

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.