The present paper concerns the linear fate of transverse perturbations in a gravity-driven, thin-film flow over a soluble substrate. We propose a reduced-order model, based on a boundary-layer treatment of the solute transport and a depth-integration of the Stokes equations, using two extended lubrication methodologies found in the literature. We obtain a closed-form dispersion relation, which we compare to a previous, fully resolved analytical investigation (Bertagni and Camporeale, J. Fluid Mech., vol. 913, 2021, A34). The results allow us to distil the essential physical mechanisms behind the instability.

‘Storm oil’ – nearly water-insoluble oil poured into the ocean and acting as a surfactant – has been used since ancient times to smooth the waves on the ocean. It was first scientifically described by Benjamin Franklin (Phil. Trans. R. Soc. Lond., vol. 64, 1774, pp. 445–460). In a recent paper, by combining highly controlled experiments in a wave tank and direct numerical simulations, Erinin et al. (J. Fluid Mech., vol. 972, 2023, R5) have now beautifully revealed the strong effect of soluble surfactants on the dynamics of plunging breakers. Remarkably, it is not the change in surface tension which mainly matters, but the surface tension gradient which emerges through compression and dilation of the plunging breaker surface.

We study the effects of buoyancy, surface-tension gradients and phase boundary on the stability of a layer of water that is confined between air at the top and a layer of ice at the bottom. The temperature of the overlying air and flux condition at the free surface of the water layer are such that the layer is susceptible to both thermal and thermocapillary instabilities. We perform a linear stability analysis to identify these modes of instability and investigate the effects of the phase boundary on them. We find that with increasing thickness of the ice layer, the critical Rayleigh and Marangoni numbers for the instabilities are found to first decrease and then asymptote to constant values for ice thicknesses much larger than the thickness of the water layer. In the case of thermocapillary instability, we find that the thickness of the ice layer has negligible influence on the stability threshold for dimensionless wavenumber $k \gg 1$, and that the presence of an unstably stratified liquid layer significantly alters the stability threshold for $k = O (1)$. Furthermore, the inclusion of Marangoni stresses reduces the stability threshold of the thermal instability.

We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, $\varepsilon$ and $\chi$. Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number $Re_b:=\varepsilon /(\nu N^2)$, where $\nu$ is the kinematic viscosity and $N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for $\varepsilon$ and $\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes $Re_b\ll 1$ and $Re_b\gg 1$, and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of $Re_b$), the PCNN outperforms assumptions of isotropy significantly as $Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.

The rheological behaviour of dense suspensions of ideally conductive particles in the presence of both electric field and shear flow is studied using large-scale numerical simulations. Under the action of an electric field, these particles are known to undergo dipolophoresis (DIP), which is the combination of two nonlinear electrokinetic phenomena: induced-charge electrophoresis (ICEP) and dielectrophoresis (DEP). For ideally conductive particles, ICEP is predominant over DEP, resulting in transient pairing dynamics. The shear viscosity and first and second normal stress differences $N_1$ and $N_2$ of such suspensions are examined over a range of volume fractions $15\,\% \leq \phi \leq 50\,\%$ as a function of Mason number $Mn$, which measures the relative importance of viscous shear stress over electrokinetic-driven stress. For $Mn < 1$ or low shear rates, the DIP is shown to dominate the dynamics, resulting in a relatively low-viscosity state. The positive $N_1$ and negative $N_2$ are observed at $\phi < 30\,\%$, which is similar to Brownian suspensions, while their signs are reversed at $\phi \ge 30\,\%$. For $Mn \ge 1$, the shear thickening starts to arise at $\phi \ge 30\,\%$, and an almost five-fold increase in viscosity occurs at $\phi = 50\,\%$. Both $N_1$ and $N_2$ are negative for $Mn \gg 1$ at all volume fractions considered. We illuminate the transition in rheological behaviours from DIP to shear dominance around $Mn = 1$ in connection to suspension microstructure and dynamics. Lastly, our findings reveal the potential use of nonlinear electrokinetics as a means of active rheology control for such suspensions.

The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears to be linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms by (i) using recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu et al., J. Fluid Mech., vol. 969, 2023, A20), (ii) averaging these DNS into two layers, and (iii) using an inviscid two-layer shallow-water and instability theory to diagnose interfacial wave behaviour and provide physical insight. The laminar flow is subcritical and stable throughout the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the flow becomes supercritical everywhere and unstable to long waves. An internal jump featuring stationary waves first appears near the centre of the duct, then leads to larger-amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence at the largest tilt angle. Long waves described by the (nonlinear) shallow-water equation are interpreted locally as linear waves on a two-layer parallel base flow described by the Taylor–Goldstein equation. This link helps us to interpret long-wave instability and contrast it with short-wave (e.g. Kelvin–Helmholtz) instability. Our results suggest a transition to turbulence in SID through long-wave instability relying on vertical confinement by the top and bottom walls.

If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power law: $h\sim t^{-1/2}$. We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex. We find that, depending upon the initial position of the plate and the position of the centre of force, the plate might either make contact in finite time or settle progressively without ever making contact. Our results show an excellent agreement between analytical exact solutions, asymptotic solutions and numerical studies of the lubrication equations.

We study turbulent flow in open channels with a free surface and rectangular cross-section, for various Reynolds numbers and duct aspect ratios. Direct numerical simulations are used to obtain accurate characterization of the secondary motions, which are found to be more intense than in closed ducts, and to scale with the bulk, rather than with the friction velocity. A notable feature of open-duct flows is the presence of a velocity dip, namely the peak velocity is achieved at some depth underneath the free surface. We find that the depth of the velocity peak increases with the Reynolds number, and correspondingly the flow becomes more symmetric with respect to the horizontal midplane. This is also confirmed from the change of the topology of the secondary motions, which exhibit a strong corner circulation at the free-surface/wall corners at low Reynolds number, which, however, weakens at higher $Re$. The structure of the mean velocity field is such that the log law applies with good approximation in the direction normal to the nearest wall, which allows us to explain why predictive friction formulae based on the hydraulic diameter concept are successful. Additional analysis shows that the secondary motions account for a large fraction of the frictional drag (up to $15$ %).

The influence of roughness spacing on boundary layer transition over distributed roughness elements is studied using direct numerical simulation and global stability analysis, and compared with isolated roughness elements at the same Reynolds number $Re_h=U_eh/\nu$ ($U_e$ is the boundary layer edge velocity, h is roughness height and $\nu$ is the kinetic viscosity of the fluid). Small spanwise spacing ($\lambda _z=2.5h$) inhibits the formation of counter-rotating vortex pairs (CVP) and, as a result, hairpin vortices are not generated and the downstream shear layer is steady. For $\lambda _z=5h$, the CVP and hairpin vortices are induced by the first row of roughness, perturbing the downstream shear layer and causing transition. The temporal periodicity of the primary hairpin vortices appears to be independent of the streamwise spacing. Distributed roughness leads to a lower critical roughness Reynolds number for instability to occur and a more significant breakdown of the boundary layer compared with isolated roughness. When the streamwise spacing is $\lambda _x=5h$, the high-momentum fluid barely moves downward into the cavities and the wake flow has little impact on the following roughness elements. The leading unstable varicose mode is associated with the central low-speed streaks along the aligned roughness elements, and its frequency is close to the shedding frequency of hairpin vortices in the isolated case. For larger streamwise spacing ($\lambda _x=10h$), two distinct modes are obtained from global stability analysis. The first mode shows varicose symmetry, corresponding to the primary hairpin vortex shedding induced by the first-row roughness. The high-speed streaks formed in the longitudinal grooves are also found to be unstable and interact with the varicose mode. The second mode is a sinuous mode with lower frequency, induced as the wake flow of the first-row roughness runs into the second row. It extracts most energy from the spanwise shear between the high- and low-speed streaks.

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number, ${\textit {Bq}}\gg 1$. A semianalytic solution of the dual integral equations governing the flow at arbitrary ${\textit {Bq}}$ was devised by Hughes et al. (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit ${\textit {Bq}}\to 0$, it produces the value $8$ for the dimensionless translational drag, which is $50\,\%$ larger than the classical $16/3$-value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit ${\textit {Bq}}\to 0$ from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction $(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$, $\gamma _E$ being the Euler–Mascheroni constant.

A cylinder immersed in a current and free to translate along a circular arc is considered to investigate the impact of path curvature on the flow-induced vibrations (FIV) occurring without structural restoring force. Path curvature magnitude ($\kappa$, inverse of path radius non-dimensionalized by the body diameter $D$) is varied from $0$ (transverse rectilinear path) to $20$, over a wide range of values of the structure to displaced fluid mass ratio, $m^\star \in [0.05,10]$. The exploration is carried out numerically at subcritical and postcritical values of the Reynolds number ($Re$, based on $D$ and the inflow velocity), i.e. below and above the critical value $47$ for the onset of flow unsteadiness when the body is fixed, up to $100$. Path curvature triggers a desynchronized regime of the flow–body system in addition to the synchronized regime typical of vortex-induced vibrations, and alters the composition of fluid forcing. The most prominent effect uncovered here is, however, a global enhancement of FIV, with three principal results: (i) vibrations and flow unsteadiness are found to arise at lower subcritical $Re$ along a curved path, down to $19.5$ versus $31$ for $\kappa =0$; (ii) the $m^\star$ range where substantial responses develop is considerably extended and encompasses the entire interval under study, which contrasts with the narrow band of low $m^\star$ identified for $\kappa =0$; (iii) the vibrations are amplified, $+45\,\%$ relative to the peak amplitude measured along a rectilinear path at $Re =100$.

We investigate jumping of sessile droplets from a solid surface in ambient oil using modulated electrowetting actuation. We focus on the case in which the electrowetting effect is activated to cause droplet spreading and then deactivated exactly at the moment the droplet reaches its maximum deformation. By systematically varying the control parameters such as the droplet radius, liquid viscosity and applied voltage, we provide detailed characterisation of the resulting behaviours including a comprehensive phase diagram separating detachment from non-detachment behaviours, as well as how the detach velocity and detach time, i.e. duration leading to detachment, depend on the control parameters. We then construct a theoretical model predicting the detachment condition using energy conservation principles. We finally validate our theoretical analysis by experimental data obtained in the explored ranges of the control parameters.

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid has a higher viscosity than the ambient fluid, thus avoiding Saffman–Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al. (J. Fluid Mech., vol. 747, 2014, pp. 218–246) studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow 1$ and $n\rightarrow 3$, when fluid flows into and out of the channel, respectively. No divergence is found in the shallow case presented here up to the breakdown of the geometric assumption. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming an inverse dependence on the ratio disagreement between the current front using the method of lines. An inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase-plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta }$, which does not show any divergent behaviour for large $n$. An asymptotic behaviour is found for $\delta$ that approaches $1/2$ for $n\gg 1$.

The influence of an applied magnetic field on the collisional plasma Richtmyer–Meshkov instability (RMI) is investigated through numerical simulation. The instability is studied within the five-moment multifluid plasma model without any simplifying assumptions such as infinite speed of light, negligible electron inertia or quasineutrality. The plasma is composed of ion and electron fluids, and elastic collisions are modelled with the Braginskii transport coefficients. A collisional regime is investigated and the magnetic field is applied in the direction of shock propagation, which is perpendicular to the density interface. The primary instability is influenced by several terms affecting the evolution of circulation, the most significant of which are the baroclinic, magnetic field torque and intraspecies collisional terms. The applied magnetic field results in a reduction of interface perturbation growth, agreeing qualitatively with previous numerical simulations for the case of an ideal multifluid plasma RMI. The only major difference in the present case's instability mitigation by applied magnetic field, relative to the ideal case with applied magnetic field, is that the elastic collisions replace and obstruct the secondary vorticity suppression mechanism through collisional dissipation of vorticity. Additionally the collisions, influenced by the combination of self-generated and the applied magnetic field, introduce anisotropy to the problem. The primary suppression mechanism for the RMI is unchanged relative to the ideal case, i.e. the magnetic field torque resisting baroclinic deposition of vorticity in the ion fluid.

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.

Local instantaneous exchanges of volume, momentum and buoyancy across turbulent/non-turbulent interfaces (TNTIs) and turbulent/turbulent interfaces (TTIs) are studied using data from direct numerical simulations of a turbulent forced fountain. We apply a novel algorithm that enables independent calculation of the instantaneous local entrainment and detrainment fluxes, and therefore, for the first time, the entrainment and detrainment coefficients according to the fountain model (Bloomfield & Kerr, J. Fluid Mech., vol. 424, 2000, pp. 197–216) are determined explicitly. Across the interface between the fountain and the ambient fluid, which is a TNTI, only volume entrainment occurs, and it is well predicted by the fountain model. Across the interface between the rising upflow and falling downflow within the fountain, which is a TTI, both entrainment and detrainment of volume, momentum and buoyancy occur – with the magnitude of both entrainment and detrainment typically being large compared with the net for all exchanges. However, the model seems to be unable to capture the momentum exchanges due to its ignorance of the pressure. We find that each conditional entrainment and detrainment rate, of volume, momentum and buoyancy, can be described accurately by Gaussian profiles, while the net exchange that is the superposition of the entrainment and detrainment cannot. Moreover, the entrainment exchange rate has its maximum closer to the fountain centreline than that of detrainment, explaining the tendency for net entrainment closer to the fountain centreline and net detrainment further away.

How locally injected turbulence spreads in space is investigated with direct numerical simulations. We consider a turbulent flow in a long triply periodic box generated by a forcing that is localized in space. The forcing is such that it does not inject any mean momentum into the flow. We show that at long times a statistically stationary state is reached where the turbulent energy density in space fluctuates around a mean profile that peaks at the forcing location and decreases fast away from it. We measure this profile as a function of the distance from the forcing region for different values of the Reynolds number. It is shown that, as the Reynolds number is increased, it converges to a Reynolds-number-independent profile, implying that turbulence spreads due to self-advection and not due to molecular diffusion. In this limit, therefore, turbulence plays the simultaneous role of cascading the energy to smaller scales and transporting it to larger distances. The two effects are shown to be of the same order of magnitude. Thus a new turbulent state is reached where turbulent transport and turbulent cascade are equally important and control its properties.

In Rayleigh–Bénard convection, the size of a flow domain and its aspect ratio $\varGamma$ (a ratio between the spatial length and height of the domain) affect the shape of the large-scale circulation. For some aspect ratios, the flow dynamics includes a three-dimensional oscillatory mode known as a jump rope vortex (JRV); however, the effects of varying aspect ratios on this mode are not well investigated. In this paper, we study these aspect ratio effects in liquid metals, for a low Prandtl number ${{Pr}}=0.03$. Direct numerical simulations and experiments are carried out for a Rayleigh number range $2.9 \times 10^4 \leq {{Ra}} \leq 1.6 \times 10^6$ and square cuboid domains with $\varGamma =2$, $2.5$, $3$ and $5$. Our study demonstrates that a repeating pattern of a JRV encountered at aspect ratio $\varGamma \approx 2.5$ is the basic structural unit that builds up to a lattice of interlaced JRVs at the largest aspect ratio. The size of the domain determines how many structural units are self-organised within the domain; the number of the realised units is expected to scale as $\varGamma ^2$ with sufficiently large and growing $\varGamma$. We find the oscillatory modes for all investigated $\varGamma$; however, they are more pronounced for $\varGamma =2.5$ and $5$. Future studies for large-aspect-ratio domains of different shapes would enhance our understanding of how the JRVs adjust and reorganise at such scaled-up geometries, and answer the question of whether they are indeed the smallest superstructure units.

We derive expressions relating the entrainment fluxes of momentum and kinetic energy, relative to the mass flux entrained into a turbulent wake exposed to a turbulent background. These expressions contain correlations between the entrainment velocity and the turbulent fluctuations within the background. We perform high-resolution, simultaneous particle image velocimetry and planar laser-induced fluorescence experiments, and observe these correlations to be negligible in the far wake, such that momentum and kinetic energy are entrained into the wake with the same relative efficiency to mass as from an idealised, non-turbulent background. This is a useful result in the context of modelling, since the entrainment hypothesis (Turner, J. Fluid Mech., vol. 173, 1986, pp. 431–471) can still be used to model the entrainment of momentum and kinetic energy. Nevertheless, the entrainment rate of mass is shown to vary spatially, and with the specific nature of the background turbulence, so this in turn drives a spatial/background-turbulence-specific entrainment rate of momentum/kinetic energy. Contrastingly, in the near wake, whilst momentum is entrained from a turbulent background with the same relative efficiency to mass as for an idealised non-turbulent background, this is not the case for kinetic energy. Owing to the sum of multiple positive, small-valued correlations between the fluctuations in the background and the entrainment velocity, kinetic energy is entrained more efficiently than in the idealised case. This includes entrainment from a non-turbulent background, where small correlations are observed between the irrotational background fluctuations and the entrainment velocity. Evidence is also presented that the entrainment velocity scales with the Kolmogorov velocity scale when the background is turbulent.

Existing experimental results show that swirling flames in annular combustors respond with a different gain to acoustic azimuthal modes rotating in either the clockwise or anti-clockwise direction. The ratio $R$ of these two gains is introduced, with $R=1$ being the conventional case of flames responding the same to the two forcing directions. To allow a difference in response to the different directions ($R\neq 1$), a multiple-input single-output azimuthal flame describing function is successfully implemented in a quaternion valued low-order model of an annular combustion chamber in the current work. Theoretical studies have explored this kind of symmetry breaking between the two acoustic wave directions in the past, but it has not been backed by experimental data. One of the main features of the new model proposed in this work is the potential difference in mode shapes between the acoustic and the heat release rate modes, which has recently been observed experimentally. This results in a gain-dependent equation for the nature of the mode, which has a significant influence on the fixed points of the system. For example, one of the spinning solutions and the standing solution can disappear through a saddle node bifurcation as the parameters are varied. The presence of only a single direction for the spinning solution matches experimental observations better than the conventional models, and the proposed model is shown to qualitatively describe experimental measurements well.