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.

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.

The Marangoni flow induced by an insoluble surfactant on a fluid–fluid interface is a fundamental problem investigated extensively due to its implications in colloid science, biology, the environment and industrial applications. Here, we study the limit of a deep liquid subphase with negligible inertia (low Reynolds number, $Re\ll {1}$), where the two-dimensional problem has been shown to be described by the complex Burgers equation. We analyse the problem through a self-similar formulation, providing further insights into its structure and revealing its universal features. Six different similarity solutions are found. One of the solutions includes surfactant diffusion, whereas the other five, which are identified through a phase-plane formalism, hold only in the limit of negligible diffusion (high surface Péclet number $Pe_s\gg {1}$). Surfactant ‘pulses’, with a locally higher concentration that spreads outward, lead to two similarity solutions of the first kind with a similarity exponent $\beta =1/2$. On the other hand, distributions that are locally depleted and flow inwards lead to similarity of the second kind, with two different exponents that we obtain exactly using stability arguments. We distinguish between ‘dimple’ solutions, where the surfactant has a quadratic minimum and $\beta =2$, from ‘hole’ solutions, where the concentration profile is flatter than quadratic and $\beta =3/2$. Each of these two cases exhibits two similarity solutions, one valid prior to a critical time $t_*$ when the derivative of the concentration is singular, and another one valid after $t_*$. We obtain all six solutions in closed form, and discuss predictions that can be extracted from these results.

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.

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.

Time-resolved two-dimensional two-component particle image velocimetry measurements with high spatial resolution are carried out in a water tank agitated by four blades rotating at constant speed. Different blade geometries and rotation speeds are tested for the purpose of modifying turbulent flow conditions. In all cases where no baffles are used to break the rotation, the Zeman length is an order of magnitude smaller than the Taylor length. Compared with the cases with baffles which break the rotation, in the unbaffled cases turbulence production and/or mean advection are significant and the turbulence nonlinearity is dramatically reduced for the horizontal (i.e. normal to the axis of rotation) two-point turbulence fluctuating velocities. This nonlinearity reduction is manifest not only in the interscale turbulent energy transfer but also in the interspace turbulent energy transfer, which nearly vanishes. However, the nonlinearity is not reduced for the vertical two-point turbulence fluctuating velocities: the corresponding interscale turbulent transfer rate is in fact intensified, and its dependence on the two-point separation distance, as well as that of the corresponding interspace turbulent transfer rate, which does not vanish, is significantly modified. Even though non-homogeneities are very different for different blades and rotation speeds in the unbaffled cases, the horizontal fluctuating velocity's second-order structure function collapses with scalings which resemble predictions for homogeneous turbulence subject to strong rotation. The vertical fluctuating velocity's second-order structure function does not collapse for different blade geometries by neither these nor the Kolmogorov predictions.

High-resolution simulations of gravity currents in the lock-exchange configuration are conducted to study the flow within the head. The simulations exhibit the geometric features of the head as reported in the laboratory experiments and numerical simulations, and provide more detailed information on the flow within the head of a gravity current. The flow in the lower part of the head, where the lobes and clefts are forming at the leading edge, is qualitatively different from but interconnected to the flow in the upper part of the head, where steepening bulges are protruding from the upright surface above the clefts. Interestingly, regions of positive and negative streamwise vorticity are observed not only in the lower part of the head but also in the upper part of the head at staggered spanwise locations. We have shown that both the streamwise vorticity at the leading edge of the lobes in the lower part of the head and the streamwise vorticity at the steepening bulges in the upper part of the head are contributed from the twisting of spanwise vorticity into the streamwise direction, due to the geometric features of the lobes and the steepening bulges, and contributed from the baroclinic production of vorticity. Our results from visualization using tracers indicate that the ambient fluid ingested in and rising from the clefts is being swept towards the leading edge of a gravity current before being carried upwards from the leading edge to the upright surface above the left and right neighbouring lobes. Furthermore, the heavy fluid inside a lobe may descend towards the bottom boundary, move forward towards the leading edge and outwards towards the neighbouring clefts, and ultimately be carried upwards to the upright surface above the left and right neighbouring lobes. With the knowledge that the erosive power of a gravity current is concentrated in the head region, it is plausible that the bed material, once resuspended by a gravity current, may be lifted up away from the bottom boundary and be dispersed in both the streamwise and spanwise directions. The present study complements existing findings in the literature and provides new insights into the three-dimensional flow field within the head of a gravity current.

An experimental study has been conducted on the near wake of a 6 : 1 spheroid, in both uniform and stratified backgrounds. The pitch angle, $\theta$, was varied from $0^\circ \text { to }20^\circ$. When $\theta = 0^\circ$, stratification decreases the characteristic wake element spacing so a characteristic Strouhal number ($St$) increases from 0.32 to 0.4. However, a similar measure scaled on wake momentum thickness shows the wake spacing to converge on those measured for other bluff and streamlined bodies. There is an apparent effect of Reynolds number, which changes the location of separation lines and hence the initial wake thickness. When $\theta > 0^\circ$, the wake is a combination of the usual drag wake together with a collection of streamwise vortices that have separated from the body, and this wake geometry can evolve in ways that are measurably different from the zero incidence case. These differences may be limited to the near wake, as the later evolution appears to converge with previous bluff- and streamlined bodies, with normalised wake height, $L_V = 0.5$ and centreline velocity, $\bar {u}_0 = 0.3$ at $Nt = 10$, as the early wake enters the non-equilibrium regime with similar values to previously studied stratified wakes. In the presence of density stratification, the inclined wake itself generates large-scale internal wave undulations with time scale $2{\rm \pi} /N$, even when the background stratification is not strong and a body-based Froude number is $O(10)$. The geometry and strengths of the primary streamwise vortices are not symmetric, mirroring previous results from experiments and computations in the literature.

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.

Trapped surface waves have been observed in a swimming pool trapped by, and rotating around, the cores of vortices. To investigate this effect, we have numerically studied the free-surface response of a Lamb–Oseen vortex to small perturbations. The fluid has finite depth but is laterally unbounded. The numerical method used is spectrally accurate, and uses a novel non-reflecting buffer region to simulate a laterally unbounded fluid. While a variety of linear waves can arise in this flow, we focus here on surface gravity waves. We investigate the linear modes of the vortex as a function of the perturbation azimuthal mode number and the vortex rotation rate. We find that at low rotation rates, linear modes decay by radiating energy to the far field, while at higher rotation rates modes become nearly neutrally stable and trapped in the vicinity of the vortex. While trapped modes have previously been seen in shallow water surface waves due to small perturbations of a bathtub vortex, the situation considered here is qualitatively different owing to the lack of an inward flow and the dispersive nature of non-shallow-water waves. We also find that for slow vortex rotation rates, trapped waves propagate in the opposite direction to the vortex rotation, whereas, above a threshold rotation rate, waves corotate with the flow.

We analyze a dataset from a numerically simulated, temporally evolving turbulent wake (Zhou, Phys. Rev. Fluids, vol. 7, 2022, 104802) that exhibits spontaneous anisotropic layering under strong stratification, alongside significant spatiotemporal variability within the flow. This analysis focuses on the irreversible flux coefficient, $\varGamma$, defined as the ratio between turbulent potential and kinetic energy dissipation rates. We find that the volume-averaged $\varGamma$ initially rises, reaches a plateau between 0.45 and 0.49 when the layering dynamics become dominant, and then decreases as viscosity plays a larger role. These peak $\varGamma$ values are consistent with those from prior simulations under strongly stratified conditions. Such efficient mixing occurs when the Ozmidov to Thorpe length scale ratio is between 0.37 and 0.52, consistent with numerical and field data reported by Mashayek et al. (J. Fluid Mech., vol. 826, 2017, pp. 522–552). To account for the coexistence of dynamically distinct regions within the flow, we perform conditional sampling of $\varGamma$ against a locally defined gradient Richardson number, ${\textit {Ri}}$. This reveals a flux-gradient relation between $\varGamma$ and ${\textit {Ri}}$ that remains largely consistent over time. This relation features a large, approximately constant value of $\varGamma$ for ${\textit {Ri}}$ values greater than one, echoing the ‘constant-power’ scenario postulated by Balmforth et al. (J. Fluid Mech., vol. 355, 1998, pp. 329–358).

The internal flow within an evaporating sessile droplet has intriguing fluid mechanics important to various microfluidics applications. In the present study, a phenomenon is observed through numerical methods wherein the buoyancy-driven flow structure inside a droplet on a non-wetting substrate transitions from an axisymmetric toroidal vortex flow to a non-axisymmetric single vortex flow with increase in the substrate temperature. As the axisymmetric nature of the droplet flow field and evaporation characteristics are broken, the internal velocity accelerates significantly. The transition, which is attributed to a flow instability inside the droplet, is more prone to occur as the droplet volume or the contact angle increases. The onset of the flow transition is analysed as the amplification of a small perturbation, thereby establishing a correlation between the flow instability and the Rayleigh number (Ra). Specifically, when Ra exceeds some critical value, the onset of the flow transition is observed, which explains the effects of substrate temperature and droplet volume on the internal flow. Next, the influence of the droplet contact angle on the critical Ra was investigated, and the underlying reasons were analysed. Finally, we discuss the heat transfer efficiency within the droplet and analyse why the internal flow tends to transition to a non-axisymmetric flow pattern from an energy minimization perspective.

To study two-dimensional dispersive waves propagating through turbulent flows, a new and less restrictive fast waves approximation is proposed using a multiscale setting. In this ansatz, large and small scales of the turbulence are treated differently. Correlation lengths of the random small-scale turbulence components can be considered negligible in the wave packet propagating frame. Nevertheless, the large-scale flow can be relatively strong, to significantly impact wavenumbers along the propagating rays. New theoretical results, numerical tools and proxies are derived to describe ray and wave action distributions. All model parameters can be calibrated robustly from the large-scale flow component only. We illustrate our purpose with ocean surface gravity waves propagating in different types of surface currents. The multiscale solution is demonstrated to efficiently document wave trapping effects by intense jets.

In studying the transport of inclusions in multiphase systems we are often interested in integrated quantities such as the net force and the net velocity of the inclusions. In the reciprocal theorem the known solution to the first and typically easier boundary value problem is used to compute the integrated quantities, such as the net force, in the second problem without the need to solve that problem. Here, we derive a reciprocal theorem for poro-viscoelastic (or biphasic) materials that are composed of a linear compressible solid phase, permeated by a viscous fluid. As an example, we analytically calculate the time-dependent net force on a rigid sphere in response to point forces applied to the elastic network and the Newtonian fluid phases of the biphasic material. We show that when the point force is applied to the fluid phase, the net force on the sphere evolves over time scales that are independent of the distance between the point force and the sphere; in comparison, when the point force is applied to the elastic phase, the time scale for force development increases quadratically with the distance, in line with the scaling of poroelastic relaxation time. Finally, we formulate and discuss how the reciprocal theorem can be applied to other areas, including (i) calculating the network slip on the sphere's surface, (ii) computing the leading-order effects of nonlinearities in the fluid and network forces and stresses, and (iii) calculating self-propulsion in biphasic systems.

The first two parts of this study showed by experiment and by large-eddy simulation that, in mixing layers formed of processions of spanwise-oriented vortex structures, there is a fundamental change in the dynamics of the large-structure growth at what has been called the ‘mixing’ transition. This third part examines the relationship of the post-transition ‘organised’ flow to the mixing layer of classical turbulence theory. Further large-eddy simulations are presented which, like some of the experiments reported in Part 1, have captured the mixing layer in both its classical and organised turbulent states, thereby allowing them to be characterised separately. The simulation results are then used to calibrate model-free integral analyses of the dynamics of the mixing layer's spatial growth. It is shown that the organised and classical flows are alternative self-preserving turbulent flow states involving fundamentally different exchanges of mass, momentum and energy, one or other of which emerges naturally in particular realisations depending on the initial conditions.

Understanding the effect of intricate surface wettability conditions on microswimmers is crucial for precisely navigating them across narrow microcirculatory networks. Here, we adopt the spherical squirmer model and Navier slip condition to delineate the microswimmer locomotion under a Poiseuille flow in a slit microchannel. Through a combined analytical–numerical approach utilizing bispherical coordinates and the superposition technique, we resolve the slip-modulated simultaneous hydrodynamic interaction with substrate boundaries. Phase portraits reveal that slip significantly alters propulsion mechanisms, destabilizing centreline stable oscillations of pullers beyond a threshold slip length. Superhydrophobic surfaces suppress near-wall rheotaxis states but preserve centreline focusing, facilitating slip-assisted directed transport without surface accumulation. Under strong background flows, subcritical Hopf bifurcation emerges for pullers at a critical slip length, transitioning dynamics from coexisting stable and unstable states to purely unstable behaviour. Contrastingly, for pushers, slip causes a transition from unstable to either stable or fixed-amplitude oscillations. Increased slip length reduces hydrodynamic repulsion on pullers from the walls by enhancing rotational velocity near the walls, whereas it counteracts the torque that causes unstable oscillations of pushers. Three-dimensional analysis of the trajectories reveals the significant role of the out-of-plane orientation of the microswimmer in its transitions between different swimming states. The presented regime maps offer parametric combinations for specific motion behaviours, guiding the development of smart microfluidic drug delivery systems and preventing biofilm deposition in biomedical devices.

The turbulent wake behind a flat-back Ahmed body is investigated using stacked stereoscopic particle image velocimetry. The wake is disturbed by a steady jet from the centre of the base and the effects are quantified for key blowing rates. The unactuated wake exhibits bistable dynamics in the horizontal plane that are completely subdued for the optimal blowing case, yielding a base drag reduction of 9 %. The three-dimensional mean wake is reconstructed and used to evaluate the wake mass fluxes whose equilibrium determines the recirculation length. The results for the unactuated wake show that up to 80 % of replenishment fluid flux entering the recirculation bubble from the free-stream flow is provided through the low-pressure side of the base, where the symmetry-breaking shear layer roll-up occurs near the base. For the optimal blowing configuration, where the wake becomes symmetric, the flux of wake replenishment is severely reduced. This flow configuration results in elongated shear layers on all sides, which terminate the bubble with a roll-up of reduced intensity at a further downstream location. The dominant cause of bubble growth and the accompanying drag reduction is attributed to the momentum of the base blowing, and the new regime is referred to as the ‘favourable momentum regime’. Similar trends are observed when the model is at $5^{\circ }$ yaw where a reduction of drag and yaw-induced asymmetry are obtained. Proper orthogonal decomposition of the wake reveals the coherent structures related to the bistable flow and the symmetric wake under optimal blowing coefficient.

The free-stream turbulence (FST) induced transition in perfect and non-ideal gas zero-pressure-gradient flat-plate boundary layers is investigated by means of large-eddy simulations. The study focuses on the influence of large incoming disturbances over the laminar-to-turbulent transition, by comparing two different integral length scales $L_f$, which differ by a factor of seven, at different FST intensities $T_u$. High-subsonic dense-gas boundary layers of the organic vapour Novec649, representative of organic Rankine cycle applications, are compared with air flows at Mach numbers 0.1 and 0.9. Compressibility and non-ideal gas effects are shown to be of minor importance in comparison to the influence of the FST integral length scale $L_f$. An increase of the inlet turbulent intensity always promotes transition, whereas an increase of $L_f$ has a double effect on the transition onset. At $T_u=2.5\,\%$, increasing $L_f$ promotes the transition, while it tends to delay transition for an FST intensity of 4 %. Larger FST integral scales tend to increase the spanwise distance between laminar streaks generated in the boundary layer. Two competing transition scenarios are observed. When the incoming turbulence intensity and length scale are moderate, the classical bypass route consists in the linear non-modal growth of streaks, which then experience secondary instabilities (sinuous or varicose) and lead to the generation of turbulent spots. The second scenario is characterized by the appearance of $\Lambda$-shaped structures near the inlet, which are further stretched to hairpin vortices before breaking down to turbulence. Spot inceptions can therefore occur at earlier locations than the streak growth. We are then faced with a competition between the classical bypass transition and nonlinear response mechanisms that ‘bypass’ this route. The present case at high $L_f$ and low $T_u$ is an example of a competing scenario, but even for the higher $T_u$ and $L_f$ conditions, only approximately one-third of turbulent spots are due to the $\Lambda$-shaped events. The nonlinear alternative route has strong similarities with scenarios described previously in the literature in the presence of leading edge effects or due to passing wakes. Such a path is governed by the turbulence intensity, but also by the integral length scale, with both parameters playing a critical role in the generation of the $\Lambda$-shaped structures near the inlet. This alternative mechanism is found to be robust under varying flow and thermodynamic conditions.

The peculiar migration and rotational dynamics of non-spherical particles in non-Newtonian flows stem from the interplay between fluid rheology and fluid inertial effects. In this paper, the cross-flow migration of a neutrally buoyant oblate spheroid (aspect ratio $AR = 0.5$) immersed in the elasto-inertial duct flow is investigated by particle-resolved simulations with the immersed boundary method. Different from spherical particles, due to the orientation-dependent lift force, the oblate spheroid migrates in an oscillating manner in the duct. The travelling period for particles reaching the duct centreline undergoes a non-monotonic change with elastic number, revealing the existence of a critical elastic number governing the migrating efficiency of oblate particles within the present flow system. For the particle rotation and orientation, the present results indicate that the particle rotation rate and orbit drifting rate are both hindered by the fluid elasticity. With increasing the fluid elasticity, three different orientation modes – log-rolling mode, kayaking-like mode and longside-flow alignment mode – are observed successively during the elasto-inertial migration of the oblate spheroid. Potentially, the present results could be used to design the rheology-based controlling strategy for guiding particles to achieve optimal focusing and orientation in microfluidic applications without the need for external forcing fields.