We report on the melting dynamics of frozen olive oil in quiescent water for Rayleigh numbers up to $10^9$. The density difference results in an upward buoyancy-driven flow of liquid oil forming a thin film around the frozen oil. We experimentally investigate flat, cylindrical and spherical shapes and we derive theoretical expressions for the local film thickness, velocity and the local melt rate for these three canonical geometries. Our theoretical models predict the correct order of magnitude and the correct scaling as compared with our experimental findings.

We consider the two-dimensional Rayleigh–Bénard convection problem between Navier-slip fixed-temperature boundary conditions, and present a new upper bound for the Nusselt number (${\textit {Nu}}$). The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result of Whitehead & Doering (Phys. Rev. Lett., vol. 106, 2011, 244501) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (Phil. Trans. R. Soc. A, vol. 380, issue 2225, 2022, 20210025) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number (${\textit {Pr}}$) and slip length, our upper bounds indicate five possible scaling laws (where ${\textit {Ra}}$ is the Rayleigh number): ${\textit {Nu}}\sim (L_s^{-1}\,{\textit {Ra}})^{{1}/{3}}$, ${\textit {Nu}}\sim (L_s^{-2/5}\,{\textit {Ra}})^{{5}/{13}}$, ${\textit {Nu}}\sim {\textit {Ra}}^{{5}/{12}}$, ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-4/3}\,{\textit {Ra}})^{{1}/{2}}$ and ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-1/3}\,{\textit {Ra}})^{{1}/{2}}$.

Recent studies of viscous dissipation mechanisms in impacting droplets have revealed distinct behaviours between the macroscale and nanoscale. However, the transition of these mechanisms from the macroscale to the nanoscale remains unexplored due to limited research at the microscale. This work addresses the gap using the many-body dissipative particle dynamics (MDPD) method. While the MDPD method omits specific atomic details, it retains crucial mesoscopic effects, making it suitable for investigating the impact dynamics at the microscale. Through the analysis of velocity contours within impacting droplets, the research identifies three primary contributors to viscous dissipation during spreading: boundary-layer viscous dissipation from shear flow; rim geometric head loss; and bulk viscous dissipation caused by droplet deformation. This prompts a re-evaluation of viscous dissipation mechanisms at both the macroscale and nanoscale. It reveals that the same three kinds of dissipation are present across all scales, differing only in their relative intensities at each scale. A model of the maximum spreading factor (βmax) incorporating all forms of viscous dissipation without adjustable parameters is developed to substantiate this insight. This model is validated against three distinct datasets representing the macroscale, microscale and nanoscale, encompassing a broad spectrum of Weber numbers, Ohnesorge numbers and contact angles. The satisfactory agreement between the model predictions and the data signifies a breakthrough in establishing a universal βmax model applicable across all scales. This model demonstrates the consistent nature of viscous dissipation mechanisms across different scales and underscores the importance of integrating microscale behaviours to understand macroscale and nanoscale phenomena.

We perform a detailed numerical study of modal and non-modal stability in oblique Couette–Poiseuille profiles, which are among the simplest examples of three-dimensional boundary layers. Through a comparison with the Orr–Sommerfeld operator for the aligned case, we show how an effective wall speed succinctly characterizes modal stability. Large-scale parameter sweeps reveal that the misalignment between the pressure gradient and wall motion is, in general, destabilizing. For flows that are sufficiently oblique, the instability is found to depend exclusively on the direction of wall motion and not on its speed, a conclusion supported, in part, by the perturbation energy budget and the evolution of the critical layers. Closed forms for the critical parameters in this regime are derived using a simple analysis. From a non-modal perspective, pseudoresonance is examined through the resolvent and the $\epsilon$-pseudospectra. An analysis of the unforced initial value problem shows that the maximum energy gain is highly dependent on both the magnitude and direction of the wall velocity. However, the strongest amplification is always achieved for configurations that are only weakly skewed. Finally, the optimal perturbations appear to develop via a lift-up effect enhanced by an Orr-like mechanism, the latter driven by cross-flow shear.

When placed at the surface of a volatile liquid, a sphere of hot dense non-volatile material remains suspended until it cools sufficiently. The duration of this ‘inverse Leidenfrost’ phenomenon depends on the Nusselt number $Nu$ of the sphere, itself determined by flow in the film of vapour separating particle and liquid. It is shown that provided the Nusselt number is large, it can be calculated numerically using only the Laplace relation and the equations governing the thin film; patching to a solution for the outer thick film is not necessary. This method is demonstrated by using it to determine $Nu$ for a sphere sufficiently small that in the governing equations, the acceleration due to gravity is negligible except where multiplied by the density of the sphere. Numerical results giving $Nu$ as a function of a dimensionless measure of sphere weight are supplemented with analysis showing that, when the weight is of the order of the maximum supportable by surface tension alone, the film consists of a spherical bubble cap bounded by its contact rim. The solutions for these regions are coupled: although the apparent contact angle $\chi$ for the cap is determined within the rim, its value depends on the flow rate arriving from the cap as well as on the additional evaporation from the rim. The latter acts to reduce $\chi$ from the value it would otherwise have, thereby reducing the thickness of the entire cap. For the example treated here, the value of $Nu$ is doubled by this mechanism.

Droplet clustering in sprays refers to the dynamic evolution of highly concentrated regions due to the preferential accumulation of the polydisperse droplets in the turbulent airflow entrained by the spray. In the current study, we aim to experimentally investigate the collective vaporization of the droplets in droplet clusters in an air-assisted acetone spray characterized by the Group number, $G$. The magnitude of $G$ depends on the cluster length scale and interdroplet spacing, and it indicates the vaporization mode that may vary from the isolated mode ($G \ll 1$) to external group mode ($G \gg 1$). The droplet measurements were obtained under atmospheric conditions at different axial and radial locations within the spray. Application of the Voronoi analysis to particle image velocimetry images of the spray droplets facilitated the identification and characterization of the droplet clusters, which allowed the measurement of $G$ for each cluster. The results highlighted that multiscale clustering of the evaporating droplets leads to multimode group evaporation of the clusters (characterized by a wide range of $G$: 0.001–10). The trend of interdroplet spacing versus cluster area allowed the classification of the droplet clusters into small-scale clusters (which are of the order of the Kolmogorov length scale) and large-scale clusters (that scale with the large-scale turbulent eddies), that are found to exhibit distinct group evaporation behaviour. A theoretical model is invoked to correlate $G$ with the droplet evaporation rate for individual clusters, and some interesting observations are identified, which are explained in the paper.

We investigate self-excited axisymmetric oscillations of a lean premixed methane–air V-flame in a laminar annular jet. The flame is anchored near the rim of the centrebody, forming an inverted cone, while the strongest vorticity is concentrated along the outer shear layer of the annular jet. Consequently, the reaction and vorticity dynamics are largely separated, except where they coalesce near the flame tip. The global eigenmodes corresponding to the linearised reacting flow equations around the steady base state are computed in an axisymmetric setting. We identify an arc branch of eigenmodes exhibiting strong oscillations at the flame tip. The associated eigenvalues are robust with respect to domain truncation and numerical discretisation, and they become destabilised as the Reynolds number increases. The frequency of the leading eigenmode is found to correspond to the Lagrangian disturbance advection time from the nozzle outlet to the flame tip. The essential role of this convective mechanism is also supported by resolvent analysis, which finds that the same flame-tip disturbance structure and frequency are optimally amplified when the flame is subjected to external white noise forcing. Strong non-modal effects in the form of pseudo-resonance are not found. Nonlinear time-resolved simulation further reveals notable hysteresis phenomena in the subcritical regime prior to instability. Hence, even when the flame is linearly stable, perturbations of sufficient amplitude can trigger limit-cycle oscillations and higher-dimensional dynamics sustained by nonlinear feedback. A Monte Carlo simulation of passive tracers in the unsteady flame suggests a nonlinear non-local instability mechanism. Notably, linear analysis of the subcritical time-averaged limit-cycle state yields eigenvalues that do not match the nonlinear periodic oscillation frequencies. This mismatch is attributed to the fundamentally nonlinear dynamics of the subcritical V-flame instability, where the dichromatic, non-local interaction between the heat release rate along the flame surface and the vortex dynamics in the jet shear layer cannot be approximated as a simple distortion of the mean flow.

We study the effects of Prandtl number $\mathit {Pr}$ and Rayleigh number $\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range $10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$, the viscous dissipation scales as $\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$, which is based on the observation that enstrophy $\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$, and the Nusselt number tends to follow the ‘ultimate’ scaling $\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of $\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum $\hat {E}_u(k) \propto k^{-2.3}$, which is close to $k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum $\hat {E}_\theta (k) \propto k^{-1.2}$, which deviates from the expected $k^{-7/5}$. Finally, at $\mathit {Pr} \to 0$ and $\infty$, we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.

Toroidal bubbles (TBs) represent cases of vortex rings with a gas–liquid interface where a gas vortex ring is encased within a liquid vortex ring, and can serve as effective media for mass conveyance, process mixing, noise reduction and reaction regulation. In this study, we carry out a systematic study on the interaction between a TB and a free surface. According to the high-speed photographic images from the experiments, we identify strong and weak interactions in terms of the normalized maximum free surface deformation $h_{max}^*$. Then, we perform numerical simulations based on the volume of fluid (VOF) method in the OpenFOAM platform. Based on both the experimental and the numerical results, we conclude that the Froude number, $Fr$, determines the main characteristics during the interaction process. The TB–free surface interaction is essentially the interaction between the liquid vortex ring enveloping the TB and the free surface, supplemented by the TB's complex behaviour. Next, we establish the scaling law of $h_{max}^*$ based on the energy balance condition. Based on this, we provide the critical $Fr$ and the slenderness of the TB, $\eta$, for identifying the strong and weak interactions, and a parametric plot of the interactions in terms of $Fr$ and $\eta$.

Craters form as the lander's exhaust interacts with the planetary surfaces. Understanding this phenomenon is imperative to ensuring safe landings. We investigate the crater morphology, where a turbulent air jet impinges on granular surfaces. To reveal the fundamental aspect of this phenomenon, systematic experiments are performed with various air-jet velocities, nozzle positions and grain properties. The resultant crater morphology is characterized by an aspect ratio. We find a universal scaling law in which the aspect ratio is scaled by a dimensionless variable consisting of the air velocity at the nozzle, the speed of sound in air, the nozzle diameter, the nozzle-tip distance from the surface, the grain diameter, the density of the grains and the density of air. The obtained scaling reveals the cross-over of the length scales governing the crater aspect ratio, providing a useful guideline for ensuring safe landings. Moreover, we report a novel drop-shaped sub-surface cratering phenomenon.

We examine the fluid flow forced by precession of a rotating cylindrical container using numerical simulations and experimental flow measurements with ultrasonic Doppler velocimetry. The analysis is based on the decomposition of the flow field into contributions with distinct azimuthal symmetry or analytically known inertial modes and the corresponding calculation of their amplitudes. We show that the predominant fraction of the kinetic energy of the precession-driven fluid flow is contained only within a few large-scale modes. The most striking observation shown by simulations and experiments is the transition from a flow dominated by large-scale structures to a more turbulent behaviour with the small-scale fluctuations becoming increasingly important. At a fixed rotation frequency (parametrized by the Reynolds number, $Re$) this transition occurs when a critical precession ratio is exceeded and consists of a two-stage collapse of the directly driven flow going along with a massive modification of the azimuthal circulation (the zonal flow) and the appearance of an axisymmetric double-roll mode limited to a narrow range of precession ratios. A similar behaviour is found in experiments which make it possible to follow the transition up to Reynolds numbers of $Re\approx 2\times 10^6$. We find that the critical precession ratio decreases with rotation, initially showing a particular scaling ${\propto }Re^{-({1}/{5})}$ but developing an asymptotic behaviour for $Re\gtrsim 10^5$ which might be explained by the onset of turbulence in boundary layers.

We investigate effect of porous insert located upstream of the separation edge of a backward-facing step (BFS) in early transitional regime as a function of Reynolds number. This is an example of hydrodynamic system that is a combination of separated shear flow with large amplification potential and porous materials known for efficient flow destabilisation. Spectral analysis reveals that dynamics of BFS is dominated by spectral modes that remain globally coherent along the streamwise direction. We detect two branches of characteristic frequencies in the flow and with Hilbert transform we characterise their spatial support. For low Reynolds numbers, the dynamics of the flow is dominated by lower frequency, whereas for sufficiently large Reynolds numbers cross-over to higher frequencies is observed. Increasing permeability of the porous insert results in decrease in Reynolds number value, at which frequency cross-over occurs. By comparing normalised frequencies on each branch with local stability analysis, we attribute Kelvin–Helmholtz and Tollmien–Schlichting instabilities to upper and lower frequency branches, respectively. Finally, our results show that porous inserts enhance Kelvin–Helmholtz instability and promote transition to oscillator-type dynamics. Specifically, the amplitude of vortical (BFS) structures associated with higher-frequency branch follows Landau model prediction for all investigated porous inserts.

We derive a generalised asymptotic model for the flow of a thin fluid film over an arbitrarily parameterised non-axisymmetric curved substrate surface based on the lubrication approximation. In addition to surface tension, gravity and centrifugal force, our model incorporates the effects of the Coriolis force and disjoining pressure, together with a non-uniform initial condition, which have not been widely considered in existing literature. We use this model to investigate the impact of the Coriolis force and fingering instability on the spreading of a non-axisymmetric spin-coated film at a range of substrate angular velocities, first on a flat substrate, and then on parabolic cylinder- and saddle-shaped curved substrates. We show that, on flat substrates, the Coriolis force has a negligible impact at low angular velocities, and at high angular velocities results in a small deflection of fingers formed at the contact line against the direction of substrate rotation. On curved substrates, we demonstrate that, as the angular velocity is increased, spin-coated films transition from being dominated by gravitational drainage with no fingering to spreading and fingering in the direction with the greatest component of centrifugal force tangent to the substrate surface. For both curved substrates and all angular velocities considered, we show that the film thickness and total wetted substrate area remain similar over time to those on a flat substrate, with the key difference being the shape of the spreading droplet.

This study aims to quantify how turbulence in a channel flow mixes momentum in the mean sense. We applied the macroscopic forcing method (Mani & Park, Phys. Rev. Fluids, 2021, 054607) to direct numerical simulation (DNS) of a turbulent channel flow at $Re_\tau =180$ using two different forcing strategies that are designed to separately assess the anisotropy and non-locality of momentum mixing. In the first strategy, the leading term of the Kramers–Moyal expansion of the eddy viscosity is quantified, revealing all 81 tensorial coefficients that essentially characterise the local-limit eddy viscosity. The results indicate the following: (1) the eddy viscosity has significant anisotropy, (2) Reynolds stresses are generated by both the mean strain rate and mean rotation rate tensors associated with the momentum field and (3) the local-limit eddy viscosity generates asymmetric Reynolds stress tensors. In the second strategy, the eddy viscosity is quantified as an integration kernel revealing the non-local influence of the mean momentum gradient at each wall-normal coordinate on all nine components of the Reynolds stresses over the channel width. Our results indicate that while the shear component of the Reynolds stress is reasonably reproduced by the local mean gradients, other components of the Reynolds stress are highly non-local. These results provide an understanding of anisotropy and non-locality requirements for closure modelling of momentum transport in attached wall-bounded turbulent flows.

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.

In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids.

Understanding the generation of large-scale magnetic fields and flows in magnetohydro-dynamical (MHD) turbulence remains one of the most challenging problems in astrophysical fluid dynamics. Although much work has been done on the kinematic generation of large-scale magnetic fields by turbulence, relatively little attention has been paid to the much more difficult problem in which fields and flows interact on an equal footing. The aim is to find conditions for long-wavelength instabilities of stationary MHD states. Here, we first revisit the formal exposition of the long-wavelength linear instability theory, showing how long-wavelength perturbations are governed by four mean field tensors; we then show how these tensors may be calculated explicitly under the ‘short-sudden’ approximation for the turbulence. For MHD states with relatively little disorder, the linear theory works well: average quantities can be readily calculated, and stability to long-wavelength perturbations determined. However, for disordered basic states, linear perturbations can grow without bound and the purely linear theory, as formulated, cannot be applied. We then address the question of whether there is a linear response for sufficiently weak mean fields and flows in a dynamical (nonlinear) evolution, where perturbations are guaranteed to be bounded. As a preliminary study, we first address the nature of the response in a series of one-dimensional maps. For the full MHD problem, we show that in certain circumstances, there is a clear linear response; however, in others, mean quantities – and hence the nature of the response – can be difficult to calculate.

We discuss the applicability of quasilinear-type approximations for a turbulent system with a large range of spatial and temporal scales. We consider a paradigm fluid system of rotating convection with vertical and horizontal temperature gradients. In particular, the interaction of rotation with the horizontal temperature gradient drives a ‘thermal wind’ shear flow whose strength is controlled by the horizontal temperature gradient. Varying this parameter therefore systematically alters the ordering of the shearing time scale, the convective time scale and the correlation time scale. We demonstrate that quasilinear-type approximations work well when the shearing time scale or the correlation time scale is sufficiently short. In all cases, the generalised quasilinear approximation systematically outperforms the quasilinear approximation. We discuss the consequences for statistical theories of turbulence interacting with mean gradients.

Regular reflection (RR) to Mach reflection (MR) transitions (${\rm RR}\leftrightarrow {\rm MR}$) on long wedges in steady supersonic flows have been well studied and documented. However, in a short wedge where the wedge length is small, the transition prediction becomes really challenging owing to the interaction of the expansion fan emanating from the trailing edge of the wedge with the incident shock and the triple/reflection point. The extent of this interaction depends on the distance between the wedge trailing edge and the symmetry line (Ht). This distance is a geometric combination of the distance of the wedge leading edge from the symmetry line $(H)$, the wedge angle ($\theta$) and the wedge length $(w)$. In the present study, we used the method of characteristics to model the complete wave interactions which accurately predicted shock curvatures and the reflection configurations for all ranges of the incoming flow Mach number. In the case of short wedges, the transition criterion strongly depends on the wedge length, which can be so adjusted even to eliminate the ${\rm RR}\rightarrow {\rm MR}$ transitions till the wedge angle reaches the no-reflection domain. Transition lines for both the detachment criterion and von Neumann criterion are also drawn to investigate the dual solution domain, and the reflection configurations were verified experimentally for the first time on short wedges. By using proper input configuration parameter ($w/H$), various types of shifts in the dual solution domain for short wedges are studied and categorised into three types, namely Type I, Type II and Type III.

We present an experimental investigation aimed at understanding the effects of surface roughness on the time-mean drag coefficient ($\bar {C}_{D}$) of finite-span cylinders ($\text {span/diameter} = \text {aspect ratio}$, $0.51 \le AR \le 6.02$) freely rolling without slip on an inclined plane. While lubrication theory predicts an infinite drag force for ideally smooth cylinders in contact with a smooth surface, experiments yield finite drag coefficients. We propose that surface roughness introduces an effective gap ($G_{eff}$) resulting in a finite drag force while allowing physical contact between the cylinder and the plane. This study combines measurements of surface roughness for both the cylinder and the plane panel to determine a total relative roughness ($\xi$) that can effectively describe $G_{eff}$ at the point of contact. It is observed that the measured $\bar {C}_{D}$ increases as $\xi$ decreases, aligning with predictions of lubrication theory. Furthermore, the measured $\bar {C}_{D}$ approximately matches combined analytical and numerical predictions for a smooth cylinder and plane when the imposed gap is approximately equal to the mean peak roughness ($R_p$) for rough cylinders, and one standard deviation peak roughness ($R_{p, 1\sigma }$) for relatively smooth cylinders. As the time-mean Reynolds number ($\overline {Re}$) increases, the influence of surface roughness on $\bar {C}_{D}$ decreases, indicating that wake drag becomes dominant at higher $\overline {Re}$. The cylinder aspect ratio ($AR$) is found to have only a minor effect on $\bar {C}_{D}$. Flow visualisations are also conducted to identify critical flow transitions and these are compared with visualisations previously obtained numerically. Variations in $\xi$ have little effect on the cylinder wake. Instead, $AR$ was observed to have a more pronounced effect on the flow structures observed. The Strouhal number ($St$) associated with the cylinder wake shedding was observed to increase with $\overline {Re}$, while demonstrating little dependence on $AR$.