We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of $10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of $0.25 \leq Pr \leq 20$. The transition is characterized by the emergence of multiple satellite eddies with increasing $Ra$, which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as $Ra_t \sim Pr^{1.41}$ where $Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

A phenomenological model is proposed to estimate the initial thickness of the liquid microlayer forming beneath a vapour bubble growing on a solid surface upon nucleate boiling. The model employs an analogy between the microlayer formation and the classic plate withdrawal problem. It calculates the microlayer thickness by considering it as a Landau–Levich film, where the thickness is a function of the meniscus speed and radius of curvature. Given the nearly hemispherical shape of the bubble during the early growth stage when the microlayer is first deposited, we assume that the meniscus speed can be approximated by the bubble expansion rate, and estimate the meniscus curvature using the Rayleigh equations. Unlike previous theories that assume that the bubble radius growth is proportional to the square root of time, the proposed model does not rely on any specific law of growth for vapour bubbles. The model is validated for predicting the microlayer thickness in water and ethanol, showing good agreement with experimental measurements and empirical correlations. Subsequent analyses of the microlayer interface profile address inconsistent reports – some described a wedge-like shape, whereas others reported a slight outward curvature with decreasing thickness in the outer region. This discrepancy is attributed to a reduction in the expansion rate of the microlayer's outer edge, particularly when the bubble reaches its maximum width. Our model provides insights into microlayer dynamics, essential to boiling heat transfer, as the evaporative heat flux through the microlayer is very sensitive to its initial thickness.

When studying two-dimensional fluid–body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free surface at every order. The waves are, in fact, exponentially small and it has been well-established that such waves ‘switch on’ seemingly instantaneously across so-called Stokes lines, partitioning the fluid domain into wave-free regions and regions with waves. In three dimensions, the Stokes-line concept extends to higher-dimensional Stokes surfaces. This article is concerned with the archetypal problem of uniform flow over a point source, reminiscent of, but separate to, the famous Kelvin wave problem. In prior research, the intersection of the Stokes surface with the free surface was found, in implicit form, for this case of a point source. However, on account of the algebraic manipulations required, it is not clear how this approach can be extended to more challenging settings. Here we develop a numerical-based procedure that allows the Stokes surface to be computed. The intersections of the Stokes surfaces with both the free surface and the deeper fluid are discussed for the case of the point source. Crucially, this procedure provides an important tool for generalising exponential asymptotics to the case of nonlinear (non-point-source) wave-generating bodies.

We use large-eddy simulations to study the penetration of a buoyant plume carrying a passive tracer into a stably stratified layer with constant buoyancy frequency. Using a buoyancy-tracer volume distribution, we develop a method for objectively partitioning plume fluid in buoyancy-tracer space into three regions, each of which corresponds to a coherent region in physical space. Specifically, we identify a source region where undiluted plume fluid enters the stratified layer, a transport region where much of the transition from undiluted to mixed fluid occurs in the plume cap and an accumulation region corresponding to a radially spreading intrusion. This method enables quantification of different measures of turbulence and mixing within each of the three regions, including potential energy and turbulent kinetic energy dissipation rates, an activity parameter and the instantaneous mixing efficiency. We find that the most intense buoyancy gradients lie in a thin layer at the cap of the penetrating plume. This provides the primary stage of mixing between plume and environment and exhibits a mixing efficiency around 50 %. Newly generated mixtures of environmental and plume fluid join the intrusion and experience relatively weak turbulence and buoyancy gradients. As the intrusion spreads radially, environmental fluid surrounding the intrusion is mixed into the intrusion with moderate mixing efficiency. This dominates the volume of environmental fluid entrained into the region containing plume fluid. However, the ‘strongest’ entrainment, as measured by the specific entrainment rate, is largest in the plume cap, where the most buoyant environmental fluid is entrained.

A viscous, lubrication-like response can be triggered in a thin film of fluid squeezed between a rigid flat surface and the tip of an incoming projectile. We develop a scaling for this viscous approach stage of fluid-mediated normal impact, applicable to soft impactors. Under the assumption of mediating fluid being incompressible, the impacting solid displays two limit regimes: one dominated by elasticity, and the other by inertia. The transition between the two is predicted by a dimensionless parameter, which can be interpreted as the ratio between two time scales that are the time that it takes for the surface waves to warn the leading edge of the impactor of the forthcoming impact, and the characteristic duration of the final viscous phase of the approach. Additionally, we elucidate why nearly incompressible solids feature (a) substantial ‘gliding’ prior to contact at the transition between regimes, (b) the largest size of entrapped bubble between the deformed tip of the impactor and the flat surface, and (c) a sudden drop in entrapped bubble radius past the transition between regimes. Finally, we argue that the above time scale ratio (a dimensionless number) can govern the different dynamics reported experimentally for a fluid droplet as a function of its viscosity and surface tension.

Numerous flying and swimming creatures use the ground effect to boost their propulsive performance, with the ‘ground’ referring to either a solid boundary or a free surface. While our knowledge of how a solid boundary affects biolocomotion is relatively comprehensive, the ground effect of a free surface is not fully understood. To address this limitation, we conduct a numerical investigation on the propulsion performance of a flapping plate under a free surface, subject to a range of control parameters. When the Froude number ($Fr$) is very low (i.e. little surface deformation), the effects of a free surface are similar to those of a solid boundary, with enhanced thrust and input power but little change in efficiency. However, as $Fr$ increases (i.e. more surface deformation), our results reveal an optimal $Fr$ of approximately 0.6, where the free surface induces a more streamlined flow around the flapping plate, effectively reducing the added mass. This results in a significant decrease in input power and greatly enhanced efficiency.

Meandering designates the main manifestation of unsteady vortex dynamics observed in experiments. This study has the twofold objective to (i) develop a theoretical model describing vortex meandering and (ii) conduct a quantitative and objective evaluation of the model against experimental data. Based on an analogy with Brownian motion, we derive the theoretical model in the framework of linear response theory. Taking the form of a Langevin equation, our model explains meandering as the competition between external excitation by free-stream perturbations, counteracted by stabilising intrinsic vortex dynamics. As such, it contains the previous approaches to explaining the phenomenon as limiting cases, and clearly highlights their shortcomings. The statistical identification of characteristic regularities in experimental data as well as the assessment of their consistency with theoretical models are important problems in physics. For samples obtained from finite-length records of correlated data, these statistical characteristics are not unique and may show spurious behaviour merely induced by the finiteness of the sample. Statistical inference provides a systematic and quantitative methodology to objectively assess the reproducibility of statistical characteristics and to evaluate their consistency with theoretical models. Their systematic application to the analysis of vortex meandering has not been done before and provides statistical evidence for our proposed Brownian-motion-like model. That is, experimental vortex meandering constitutes the manifestation of a stationary Gauss–Markov random process, which implies that the dynamics admits an ergodic probability measure.

We analyse the effect of drop-deformation-induced change in streamline topology on the scalar transport rate (the Nusselt number $Nu$) in an ambient planar linear flow. The drop-phase resistance is assumed dominant, and the drop deformation is characterised by the capillary number ($Ca$). For a spherical drop ($Ca = 0$) in an ambient planar extension, closed streamlines lead to $Nu$ increasing with the Péclet number ($Pe$), from $Nu_0$, corresponding to purely diffusive transport, to $4.1Nu_0$, corresponding to a large-$Pe$ diffusion-limited plateau. For non-zero $Ca$, we show that the flow field consists of spiralling streamlines densely wound around nested tori foliating the deformed drop interior. Now $Nu$ increases beyond the aforementioned primary plateau, saturating in a secondary one that approaches $22.3Nu_0$ for $Ca \rightarrow 0$, $Pe\,Ca \rightarrow \infty$. The enhancement appears independent of the drop-to-medium viscosity ratio. We further show that this singular dependence, of the transport rate on drop deformation, is generic across planar linear flows; chaotically wandering streamlines in some of these cases may even lead to a tertiary enhancement regime.

The elasto-inertial focusing and rotating characteristics of spheroids in a square channel flow of Oldroyd-B viscoelastic fluids are studied by the direct forcing/fictitious domain method. The rotational behaviours, changes in the equilibrium positions and travel distances are explored to analyse the mechanisms of spheroid migration in viscoelastic fluids. Within the present simulated parameters (1 ≤ Re ≤ 100, 0 ≤ Wi ≤ 2, 0.4 ≤ α ≤3), the results show that there are four kinds of equilibrium positions and six (five) kinds of rotational behaviours for the elasto-inertial migration of prolate (oblate) spheroids. We are the first to identify a new rotational mode for the migration of prolate spheroids. Only when the particles are initially located at a corner and wall bisector, some special initial orientations of the spheroids have an impact on the final equilibrium position and rotational mode. In other general initial positions, the initial orientation of the spheroid has a negligible effect. A higher Weissenberg number means the faster the particles migrate to the equilibrium position. The spheroid gradually changes from the corner (CO), channel centreline (CC), diagonal line (DL) and cross-section midline (CSM) equilibrium positions as the elastic number decreases, depending on the aspect ratio, initial orientation and rotational behaviour of the particles and the elastic number of the fluid. When the elastic number is less than the critical value, the types of rotational modes of the spheroids are reduced. By controlling the elastic number near the critical value, spheroids with different aspect ratios can be efficiently separated.

Internal solitary waves are a widely observed phenomenon in natural waters. Mathematically, they are fundamentally a nonlinear phenomenon that differs from the paradigm of turbulence, in that energy does not move across scales. Internal solitary waves may be computed from the Dubreil–Jacotin Long equation, which is a scalar partial differential equation that is equivalent to the stratified Euler equations. When a background shear current is present the algebraic complexity of the problem increases substantially. We present an alternative point of view for characterizing the situation with a shear current using Lagrangian (particle-like) models analysed with graph theoretic methods. We find that this yields a novel, data-centric framework for analysis that could prove useful well beyond the study of internal solitary waves.

Predicting the transient flow fields that develop when a shock wave passes through an area expansion is a fundamental problem in compressible fluid mechanics and significant in many engineering applications. Experiments, large eddy simulations and geometrical shock dynamics are used to study the mechanism by which a normal shock wave that expands across an area expansion evolves into a uniform normal shock far downstream of it. This study analyses shock waves with moderate Mach numbers of 1.1–1.8 that expand at area ratios of up to 5. As the shock wave propagates into the expanded region, it experiences rapid deceleration, forming a non-uniform shock front. Impinging on the walls of the larger cross-section region, the shock wave reflects and generates a complex and highly transient shock pattern near the expansion region. We have found that as the shock front propagates further downstream, a laterally moving shock wave that intersects the shock front at a triple point reverberates laterally between the walls. This process effectively evens out the flow behind the incident shock front, thus reducing the variation of properties behind it. The extended duration of this process leads to significant pressure fluctuation behind the shock front. The results show that the evolution of the shock front can be scaled using the expanded region height and the velocity of the shock wave far downstream of the expansion. The results enabled the formulation of a simple empirical relation, allowing us to predict the shock velocity far downstream of gradual and abrupt area expansions.

Research on the settling dynamics of snow particles, considering their complex morphologies and real atmospheric conditions, remains scarce despite extensive simulations and laboratory studies. Our study bridges this gap through a comprehensive field investigation into the three-dimensional (3-D) snow settling dynamics under weak atmospheric turbulence, enabled by a 3-D particle tracking velocimetry (PTV) system to record over a million trajectories, coupled with a snow particle analyser for simultaneous aerodynamic property characterization of four distinct snow types (aggregates, graupels, dendrites, needles). Our findings indicate that while the terminal velocity predicted by the aerodynamic model aligns well with the PTV-measured settling velocity for graupels, significant discrepancies arise for non-spherical particles, particularly dendrites, which exhibit higher drag coefficients than predicted. Qualitative observations of the 3-D settling trajectories highlight pronounced meandering in aggregates and dendrites, in contrast to the subtler meandering observed in needles and graupels, attributable to their smaller frontal areas. This meandering in aggregates and dendrites occurs at lower frequencies compared with that of graupels. Further quantification of trajectory acceleration and curvature suggests that the meandering frequencies in aggregates and dendrites are smaller than that of morphology-induced vortex shedding of disks, likely due to their rotational inertia, and those of graupels align with the small-scale atmospheric turbulence. Moreover, our analysis of vertical acceleration along trajectories elucidates that the orientation changes in dendrites and aggregates enhance their settling velocity. Such insights into settling dynamics refine models of snow settling velocity under weak atmospheric turbulence, with broader implications for more accurately predicting ground snow accumulation.

We investigate the dynamics of a columnar Taylor–Green vortex array under strong stratification, focusing on Froude numbers $0.125\leq Fr \leq 1.0$, with the aim of identifying and understanding the primary instabilities that lead to the vortices’ breakdown. Linear stability analysis reveals that the fastest-growing vertical wavenumber scales with $Fr^{-1}$, while the dimensionless growth rate remains approximately constant. The most unstable eigenmode, identified as the mixed hyperbolic mode by Hattori et al. (J. Fluid Mech., vol. 909, 2021, A4), bears significant similarities to the zigzag instability, first discovered by Billant & Chomaz (J. Fluid Mech., vol. 418, 2000, pp. 167–188). Direct numerical simulations further confirm that the zigzag instability is crucial in amplifying initial random perturbations to finite amplitude, with the flow structure and modal growth rate consistent with the linear stability analysis. In particular, the characteristic vertical length scale of turbulence matches that of the fastest-growing linear mode. These findings underscore the broader relevance of the zigzag instability mechanism beyond its initial discovery in vortex pairs, demonstrating its role in facilitating direct energy transfer from vertically uniform vortical motions to a characteristic vertical length scale proportional to $Fr$ in strongly stratified flows.

We propose a simple method to identify unstable parameter regions in general inviscid unidirectional shear flow stability problems. The theory is applicable to a wide range of basic flows, including those that are non-monotonic. We illustrate the method using a model of Jupiter's alternating jet streams based on the quasi-geostrophic equation. The main result is that the flow is unstable if there is an interval in the flow domain for which the reciprocal Rossby Mach number (a quantity defined in terms of the zonal flow and potential vorticity distribution), surpasses a certain threshold or ‘hurdle’. The hurdle height approaches unity when we can take the hurdle width to greatly exceed the atmosphere's intrinsic deformation length, as holds on gas giants. In this case, the Kelvin–Arnol’d sufficient condition of stability accurately detects instability. These results improve the theoretical framework for explaining the stable maintenance of Jupiter and Saturn's jets over decadal time scales.

We numerically study the influence of a soluble surfactant on the microjetting mode of the liquid–liquid flow focusing configuration. The surfactant adsorbs on the interface next to the feeding capillary and accumulates in front of the emitted jet, significantly lowering the surface tension there. The resulting Marangoni stress substantially alters the balance of the tangential stresses at the interface but does not modify the interface velocity. The global stability analysis at the minimum flow rate stability limit shows that the Marangoni stress collaborates with soluto-capillarity to stabilize the microjetting mode. Our analysis unveils the noticeable effect of the Marangoni stress associated with the surface tension perturbation. Surfactant diffusion and desorption hardly affect the stability limit. Transient numerical simulations show how subcritical and supercritical base flows respond to a spatially localized initial perturbation. Our parametric study indicates that the minimum flow rate ratio depends on the adsorption constant and the surfactant concentration through the product of these two variables. The surfactant stabilizing effect increases with the outer stream flow rate. We show that surfactants not only stabilize the microemulsion resulting from the jet breakup in hydrodynamic focusing, but also allow for the reduction of droplet size. Our findings advance the fundamental understanding of the complex role of surfactants in tip streaming via hydrodynamic focusing. In particular, our results contradict the common assumption that adding surfactant favours tip streaming simply because it reduces the meniscus tip surface tension.

In this paper it is shown that a modal detuned instability of periodic near-wall streaks originates a large-scale structure in the bulk of the turbulent channel flow. The effect of incoherent turbulent fluctuations is included in the linear operator by means of an eddy viscosity. The base flow is an array of periodic two-dimensional streaks, extracted from numerical simulations in small domains, superposed to the turbulent mean profile. The stability problem for a large number of periodic units is efficiently solved using the block-circulant matrix method proposed by Schmid et al. (Phys. Rev. Fluids, vol. 2, 2017, 113902). For friction Reynolds numbers equal or higher than $590$, it is shown that an unstable branch is present in the eigenspectra. The most unstable eigenmodes display large-scale modulations whose characteristic wavelengths are compatible with the large-scale end of the premultiplied velocity fluctuation spectra reported in previous computational studies. The wall-normal location of the large-wavelength near-wall peak in the spanwise spectrum of the eigenmode exhibits a power-law dependence on the friction Reynolds number, similarly to that found in experiments of pipes and boundary layers. Lastly, the shape of the eigenmode in the streamwise-wall-normal plane is reminiscent of the superstructures reported in the recent experiments of Deshpande et al. (J. Fluid Mech., vol. 969, 2023, A10). Therefore, there is evidence that such large-wavelength instabilities generate large-scale motions in wall-bounded turbulent flows.