We found a multi-scale steady solution of the Boussinesq equations for Rayleigh–Bénard convection in a three-dimensional periodic domain between horizontal plates with a constant temperature difference. This was realised using a homotopy from the wall-to-wall optimal transport solution provided by Motoki et al. (J. Fluid Mech., vol. 851, 2018, R4). A connected steady solution, which is a consequence of bifurcation from a thermal conduction state at Rayleigh number $Ra\sim 10^{3}$, is tracked up to $Ra\sim 10^{7}$ using a Newton–Krylov iteration. The three-dimensional exact coherent thermal convection exhibits a scaling of $Nu\sim Ra^{0.31}$ (where $Nu$ is the Nusselt number) as well as multi-scale thermal plume and vortex structures, which are quite similar to those in turbulent Rayleigh–Bénard convection. The mean temperature profiles and the root-mean-square of the temperature and velocity fluctuations are in good agreement with those of the turbulent states. Furthermore, the energy spectrum follows Kolmogorov's $-5/3$ scaling law with a consistent prefactor, and the energy transfer to small scales with a nearly constant flux in the wavenumber space is in accordance with the turbulent energy transfer.

Transient electrohydrodynamics of a liquid drop at finite Reynolds numbers is studied by computer simulations. The governing equations of the problem are solved using a parallelized front tracking/finite difference method in the framework of the Taylor–Melcher leaky dielectric theory. The effect of the dielectric properties of the fluids on the dynamic response of the drop is studied by considering three representative fluid systems, which encompass all the regions of the deformation–circulation map. A parametric study is performed over a range of capillary number $Ca$, the Ohnesorge number squared $Oh^{2}$ and the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$ to explore the effect of these parameters on the dynamic response. At low to intermediate capillary numbers, the dynamic response resembles that of a first- or second-order mechanical system, where the deformation settles to a steady state monotonically or through oscillations. However, beyond a threshold capillary number, the dynamic response becomes nonlinear. It is shown that the Ohnesorge number squared $Oh^{2}$ and the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$ are the key parameters that determine the transition from a monotonic response to an oscillatory one (and vice versa) in density-matched fluid systems. To predict the dynamic response of nearly spherical drops at arbitrary $Oh^{2}$, an analytical equation is developed, which results in a critical Ohnesorge number squared $Oh_{c}^{2}$ that demarks a monotonic response from an oscillatory one. Investigations on the effect of the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$ on the dynamic response shows that the relaxation time (for monotonic response) increases with an increase in $\widetilde{\unicode[STIX]{x1D707}}$ and that the steady state deformation ${\mathcal{D}}_{ss}$ correlates positively with $\widetilde{\unicode[STIX]{x1D707}}$ for oblate drops; however, depending on the corresponding regions of prolate drops on the deformation–circulation map, ${\mathcal{D}}_{ss}$ correlates positively or negatively with $\widetilde{\unicode[STIX]{x1D707}}$. A long-standing misconception regarding the oblate region is also rectified by development of a normal stress map, which determines the relative importance of the electric pressure and the normal hydrodynamic stress in setting the sense of the deformation of the drop.

Supercritical ${\rm CO}_2$ injection and dissolution into deep brine aquifers allow its sequestration within geological formations. After injection, ${\rm CO}_{2}$ gas phase is buoyancy-driven over the denser aqueous brine, reaching an apparent gravitational stable distribution. However, ${\rm CO}_2$ dissolution in brine propels convection since the mixture is even denser than the underlying brine. This process still needs to be characterised comprehensively. Here, we investigate the irreversible mixing of dissolved ${\rm CO}_2$ in brine through laboratory-scale numerical experiments utilising the Hele-Shaw model (Letelier et al., J. Fluid Mech., vol. 864, 2019, pp. 746–767) and a fully miscible two-fluid system. In this scenario, mixing the less dense fluid – mimicking ${\rm CO}_{2}$ gas phase – with the heavier fluid – representing aqueous brine – catalyses cabbeling-powered convection. Our numerical simulations recover the laboratory results in porous media by Neufeld et al. (Geophys. Res. Lett., vol. 37, issue 22, 2010, L22404) and may explain the scaling law obtained by Backhaus et al. (Phys. Rev. Lett., vol. 106, issue 10, 2011, 104501) in Hele-Shaw cells. More remarkably, we show that the mass flux between the two analogue fluids, characterised by the Sherwood number $ {{Sh}}$, obeys the universal scaling law $ {{Sh}}\sim {{Ra}}\, \vartheta _{scalar}$, with $ {{Ra}}$ the Rayleigh number and $\vartheta _{scalar}$ the mean scalar dissipation rate. This paper sheds light on the fluid dynamics and solubility trapping in geological carbon sequestration.

It has been reported that the follower in a tandem configuration with no wall (0W) reduces the time-averaged input power by utilizing the vortex interception mode (Zhu et al., Phys. Rev. Lett., vol. 113, 2014, p. 238105). In the present study, a numerical simulation is conducted with two self-propelled flexible fins in the tandem configuration near a single wall (1W) and two parallel walls (2W). Contrary to the vortex interception for 0W, the follower employs spontaneously a mixed mode (i.e. a combination of the vortex interception mode and the slalom mode) for 1W and the slalom mode for 2W. Although the lateral motion of the follower for 0W, 1W and 2W is synchronized with the induced lateral flow generated by the leader, the time-averaged input power of the follower for 1W and 2W is reduced significantly due to the enhanced lateral flow by the vortex–vortex interaction near the wall. The jet-like flow opposite to the moving direction continuously hinders the movement of the follower for 0W, whereas the follower for 1W and 2W utilizes the negative horizontal flow when passing between the main vortex and the induced vortex near the wall, leading to a decrease of the thrust force acting on the follower allowing the follower to keep pace with the leader. The global efficiency of the schooling fins is optimized with a small heaving amplitude of the follower and a critical value of phase difference between the leader and follower when the values of the wall proximity and bending rigidity are moderate.

This study presents computational simulations of multicomponent and multiphase flows to reproduce the physical phenomena in the secondary atomization of a droplet induced by a hot temperature environment. The computational fluid dynamics model is based on the geometric volume of fluid method, with piecewise linear interface calculation reconstruction for accurate determination of the curvature and evaporation fluxes at the interface. The purpose of the model was to faithfully reproduce complex physical processes, such as internal gas cavity formation, liquid–vapour interface instability, cavity collapse and liquid jet ejection, and the pinch-off of a secondary droplet, leading to the microexplosion phenomenon that greatly enhances the evaporation rate of non-volatile liquid droplets. The solver was validated against the analytical solution in benchmark cases, and experimental data with bicomponent droplets reported in the literature. The developed model was used to predict the atomization of heavy fuel oil exposed at high temperatures under microgravity conditions. Different atomization regimes were identified, depending on the initial size of the internal bubbles. While small bubbles led to simple gas ejections, cavity collapse caused the larger bubbles to produce a jet formation. When the ratio between the bubble and droplet volumes was bigger than 0.7, microexplosions occurred. The results were found to be consistent with cases of bubble burst on flat surfaces, showing a strong dependence on the Ohnesorge number ($Oh$). Key observable quantities, particularly jet velocity and bubble cap drainage velocity, were found to agree with correlations reported in other studies. The similarities were also supported by studies extending over a wide range of simulations (4000 cases) at different $Oh$. An inversion in the dependence of the jet velocity on $Oh$ (above a critical value $Oh_c$) was observed.

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this JFM Perspectives article a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of $n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.

We numerically investigate the mechanism resulting in fate change of a hydrophobic sphere impacting onto a confined pool, that is, at the same impact speed, it does not submerge in a wide pool but does in a narrow pool. We find that the reflection of the impact-induced gravity-capillary waves from the pool boundary is responsible for this phenomenon. In particular, the return of the wave to the symmetry axis may coincide with the rising of the impacting sphere to the water surface, which corresponds to the critical conditions of the fate change. Moreover, for the spheres at the onset of submersion in a wide pool, our analysis suggests that this scenario also accounts for an interesting observation in the numerical simulations. That is, the effective pool size $S_{c}$, beyond which the submersion of impacting spheres is no longer affected by the pool size $S$, is mainly dependent on the sphere diameter, no matter whether the surface waves are the capillary or gravity waves. For $S<S_{c}$, two important pool sizes ($S_{w,1}$ and $S_{w,2}$, and $S_{w,2}\geqslant S_{w,1}$) are identified at the same impact speed, and the sphere submersion takes place at $S_{w,1}\leqslant S\leqslant S_{w,2}$. Based on the flow features identified in simulations, a scaling law is proposed to correlate the Weber number and Bond number with $S_{w}$. The theoretical prediction is shown to agree well with the numerical results.

This study explores the impact of small-scale variability in the bottom relief on the dynamics and evolution of broad baroclinic flows in the ocean. The analytical model presented here generalizes the previously reported barotropic ‘sandpaper’ theory of flow–topography interaction to density-stratified systems. The multiscale asymptotic analysis leads to an explicit representation of the large-scale effects of irregular bottom roughness. The utility of the multiscale model is demonstrated by applying it to the problem of topography-induced spin-down of an axisymmetric vortex. We find that bathymetry affects vortices by suppressing circulation in their deep regions. As a result, vortices located above rough topography tend to be more stable than their flat-bottom counterparts. The multiscale theory is validated by comparing corresponding topography-resolving and parametric simulations.

A kinetic model called the $\nu$-model is proposed to replace the complicated Boltzmann collision operator in the simulation of rarefied flows of monatomic gas. The model follows the relaxation-time approximation, but the collision frequency (i.e. inverse relaxation time) is a function of the molecular velocity to reflect part of the collision details of the Boltzmann equation, and the target velocity distribution function (VDF) to which the VDF relaxes is close to that used in the Shakhov model. Based on the numerical simulation of strong non-equilibrium shock waves, a half-theoretical and half-empirical collision frequency is designed for different intermolecular potentials: the $\nu$-model shows significantly improved accuracy, and the underlying mechanism is analysed. The $\nu$-model also performs well in canonical rarefied microflows, especially in thermal transpiration, where kinetic models with velocity-independent collision frequency lack the capability to distinguish the influence of intermolecular potentials.

In this paper, we develop an asymptotic theory for steadily travelling gravity–capillary waves under the small-surface tension limit. In an accompanying work (Shelton et al., J. Fluid Mech., vol. 922, 2021) it was demonstrated that solutions associated with a perturbation about a leading-order gravity wave (a Stokes wave) contain surface-tension-driven parasitic ripples with an exponentially small amplitude. Thus, a naive Poincaré expansion is insufficient for their description. Here, we develop specialised methodologies in exponential asymptotics for derivation of the parasitic ripples on periodic domains. The ripples are shown to arise in conjunction with Stokes lines and the Stokes phenomenon. The resultant analysis associates the production of parasitic ripples to the complex-valued singularities associated with the crest of a steep Stokes wave. A solvability condition is derived, showing that solutions of this type do not exist at certain values of the Bond number. The asymptotic results are compared with full numerical solutions and show excellent agreement. The work provides corrections and insight of a seminal theory on parasitic capillary waves first proposed by Longuet-Higgins (J. Fluid Mech., vol. 16, issue 1, 1963, pp. 138–159).

The effects of dispersion acting on gravity currents propagating through porous media are considered theoretically and experimentally. We exploit the large aspect ratio of these currents to formulate a depth-averaged model of the evolution of the mass and buoyancy. Dispersion, acting predominantly at the interface between the current and the ambient, is velocity dependent and acts to entrain fluid into the gravity current, in direct analogy to turbulent mixing. Here, we show that when the gravity current is fed by a constant buoyancy and mass flux the buoyancy of the current is self-similar and recovers the classical solution for gravity currents in porous media. In contrast, the profile and the depth-averaged concentration of the current evolve in a non-self-similar manner. The total volume of the current increases with time as $t^{1/3}$ due to this dispersive entrainment. We test our theoretical predictions using a suite of laboratory experiments in which the evolution of the concentration within the current was mapped using a dye-attenuation technique. These experimental results show good agreement with the early-time limits of our theoretical model, and in particular accurately predict the evolution of the depth-averaged concentration profile. These results suggest that mixing within porous media may be modelled using an effective dispersive entrainment, the magnitude of which may be set by the underlying structure of the porous medium.

Rewetting is the establishment of water–surface contact that occurs during quenching of high temperature surfaces by water jet impingement. Rewetting is an unexpectedly complex phenomenon that has been reported to occur at surface temperatures significantly higher than the superheating limit of water. The presence of intermittently wet and dry episodes, and in particular the occurrence of so-called explosive boiling, is one of the theories to explain the contact of water with high temperature surfaces. However, there is a lack of experimental data in the literature to prove the presence of explosive boiling and intermittent wetting due to the small duration and scale of the rewetting phenomenon. In this study, recordings of the jet stagnation zone during rewetting are provided at a frame rate of 81 kfps. The high-speed recordings show a flashing regime consisting of intermittent (dry) bubble-rich and (wet) bubble-free periods at frequencies up to 40 kHz when the rewetted surface temperature exceeds the water superheat limit. As far as the authors know, these are the first direct observations of intermittent dry–wet periods occurring in the jet stagnation zone during quenching by water jet impingement. The dependency of the flashing frequency on initial surface temperature is quantified. A correlation between the size of the rewetting patch and the flashing frequency is found. Finally, a hypothesis to explain the role of water subcooling in maintaining the water–surface contact at surface temperatures well above the superheating limit of water is presented.

The experimental and numerical investigation presented by Bertsch et al. (Phys. Rev. Fluids, vol. 5, 2020a, p. 054202) describes the self-sustained oscillations induced by the interaction of two impinging jets in microfluidic devices. While the oscillatory regime induced by interacting jets has been studied in detail, the physical mechanism behind these oscillations remains still undetermined. The present paper focuses on two-dimensional oscillators subjected to a fully developed inlet flow, as in Bertsch et al. (Phys. Rev. Fluids, vol. 5, 2020a, p. 054202) and in contradistinction with Pawlowski et al. (J. Fluid Mech., vol. 551, 2006, pp. 117–139), who focused on plug inlet flow. The linear global stability analysis performed confirms the existence of an oscillating global mode, whose spatial structure qualitatively coincides with that computed numerically by Bertsch et al. (Phys. Rev. Fluids, vol. 5, 2020a, p. 054202), suggesting that the physical mechanism from which the oscillations would originate is predominantly two-dimensional. The interaction of the oscillating mode with a steady symmetry-breaking mode is examined making use of the weakly nonlinear theory, which shows how the system exhibits hysteresis in a certain range of aspect ratios. Lastly, sensitivity analysis is exploited to identify the wavemaker associated with the global modes, whose examination allows us to spot the core of the symmetry-breaking instability at the stagnation point and to identify the Kelvin–Helmholtz instability, located in the jets interaction region, as the main candidate for the origin of the oscillations observed in fluidic devices.