The wetting effect has attracted great scientific interest because of its natural significance as well as technical applications. Previous models mostly focus on one-component fluids or binary immiscible liquid mixtures. Modelling of the wetting phenomenon for multicomponent and multiphase fluids is a knotty issue. In this work, we present a thermodynamically consistent diffuse interface model to describe the wetting effect for ternary fluids, as an extension of Cahn's theory for binary fluids. In particular, we consider both immiscible and miscible ternary fluids. For miscible fluids, we validate the equilibrium contact angle and the thermodynamic pressure with Young's law and the Young–Laplace equation, respectively. Distinct flow patterns for dynamic wetting are presented when the surface tension and the viscous force dominate the wetting effect. For immiscible ternary fluids, we manipulate the wettability of two contact droplets deposited on a solid substrate according to three scenarios: (I) both droplets are hydrophilic; (II) a hydrophilic droplet in contact with a hydrophobic one; (III) both droplets are hydrophobic. The contact angles at each triple junction from the simulations are compared with Young's contact angle and Neumann's triangle rule. Simulations for the validation of our work are performed in two and three dimensions. In addition, we model the evaporation process of a ternary droplet and obtain the same power law as that of previous experiments. Our model allows one to relate the interfacial energies with surface composition, enabling the modelling of the coffee-ring phenomenon in further perspective.

We use experiments to explore the effect of surfactants on bubble-induced turbulence (BIT) at different scales, considering how the bubbles affect the flow kinetic energy, anisotropy and extreme events. To this end, high-resolution particle shadow velocimetry measurements are carried out in a bubble column in which the flow is generated by bubble swarms rising in water for two different bubble diameters (3 and 4 mm) and moderate gas volume fractions (0.5 %–1.3 %). We use tap water as the base liquid and add 1-Pentanol as an additional surfactant with varying bulk concentration, leading to different bubble shapes and surface boundary conditions. The results reveal that with increasing surfactant concentration, the BIT generated increases in strength, even though bubbles of a given size rise more slowly with surfactants. We also find that the level of anisotropy in the flow is enhanced with increasing surfactant concentration for bubbles of the same size, and that for the same surfactant concentration, smaller bubbles generate stronger anisotropy in the flow. Concerning the intermittency quantified by the normalized probability density functions of the fluid velocity increments, our results indicate that extreme values in the velocity increments become more probable with decreasing surfactant concentration for cases with smaller bubbles and low gas void fraction, while the effect of the surfactant is much weaker for cases with larger bubble and higher void fractions.

This paper reports several new classes of unstable recurrent solutions of the two-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions are in many ways analogous to recurrent solutions of the Navier–Stokes equation which are often referred to as exact coherent structures. In particular, we find that recurrent solutions of the Euler equation are dynamically relevant: they faithfully reproduce large-scale flows in simulations of turbulence at very high Reynolds numbers. On the other hand, these solutions have a number of properties which distinguish them from their Navier–Stokes counterparts. First of all, recurrent solutions of the Euler equation come in infinite-dimensional continuous families. Second, solutions of different types are connected, e.g. an equilibrium can be smoothly continued to a travelling wave or a time-periodic state. Third, and most important, they are only weakly unstable and, as a result, fully developed turbulence mimics some of these solutions remarkably frequently and over unexpectedly long temporal intervals.

The mean flow behaviour of a turbulent boundary layer over rough walls is expected to exhibit symmetries that govern the flow dynamics. In particular, when roughness elements are arranged in a spanwise symmetric manner, the mean flow above them should also exhibit spanwise symmetry. This symmetrical consideration has garnered substantial empirical support. We conduct direct numerical simulations (DNS) of flow over aligned cube arrays to test such symmetry considerations further. We vary the surface coverage density from 0.25 % to 6.25 %, and employ an averaging time of about 100 large-eddy turnover times, which is longer than the typical averaging time in prior DNS studies of rough-wall boundary layers. The results suggest the presence of spanwise asymmetry in the mean flow. Specifically, we observe the development of a prominent secondary vortex on one side of the cubical roughness, accompanied by a relatively smaller secondary vortex on the other side. This asymmetry becomes most pronounced when the surface coverage density is approximately 0.59 %, and diminishes as the coverage density approaches either a low or a high value. We also establish that this mean flow asymmetry is robust across variations in the domain size, the initial condition, and the placement of the cubes in the spanwise direction.

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

This work discusses modons, or dipolar vortices, propagating along sloping topography. Two different regimes exist, which are studied separately using the surface quasi-geostrophic equations. First, when the modon propagates in the direction opposite to topographic Rossby waves, steady solutions exist and a semi-analytical method is presented for calculating these solutions. Second, when the modon propagates in the same direction as the Rossby waves, a wave wake is generated. This wake removes energy from the modon, causing it to decay slowly. Asymptotic predictions are presented for this decay and found to agree closely with numerical simulations. Over long times, decaying vortices are found to break down due to an asymmetry resulting from the generation of waves inside the vortex. A monopolar vortex moving along a wall is shown to behave in a similar way to a dipole, though the presence of the wall is found to stabilise the vortex and prevent the long-time breakdown. The problem is equivalent mathematically to a dipolar vortex moving along a density front, hence our results apply directly to this case.

We present novel numerical simulations investigating the bag breakup of liquid droplets. We first examine the viscous effect on the early-time drop deformation, comparing with theory and experiment. Next, a bag film forms at late time and is susceptible to spurious mesh-induced breakup in numerical simulations, which has prevented previous studies from reaching grid convergence of fragment statistics. We therefore adopt the manifold death (MD) algorithm which artificially perforates thin films once they reach a prescribed critical thickness independent of the grid size, controlled by a numerical parameter $L_{sig}$. We show grid convergence of fragment statistics when utilising the MD algorithm, and analyse the fragment behaviour and bag film disintegration mechanisms including ligament breakup, node detachment and rim destabilisation. Our choice of the critical thickness parameter $L_{sig}$ is limited by numerical constraints and thus has not been matched to experiment or theory; consequently, the current simulations yield critical bag film perforation thicknesses larger than experimentally observed. The influence of the MD algorithm configuration on the bag breakup phenomena and statistics will be investigated in future work. We also study the effects of moderate liquid Ohnesorge number ($0.005 \leqslant Oh \leqslant 0.05$) on the bag breakup process and fragment statistics, where a non-monotonic dependency of the average diameter of bag film fragments on $Oh$ is found. These results highlight the utility of the MD algorithm in multiphase simulations involving topological changes, and pave the way for physics-based numerical investigations into spume generation at the air–sea interface.

A mathematical model for the effect of the spatial variation of the local evaporative flux on the evaporation of and deposition from a thin pinned particle-laden sessile droplet is formulated and solved. We then analyse the behaviour for a one-parameter family of local evaporative fluxes with the free parameter $n \, (>-1)$ that exhibits qualitatively different behaviours mimicking those that can be obtained by, for example, surrounding the droplet with a bath of fluid or using a mask with one or more holes in it to achieve a desired pattern of evaporation enhancement and/or suppression. We show that when $-1< n<1$ (including the special cases $n=-1/2$ of diffusion-limited evaporation into an unbounded atmosphere and $n=0$ of spatially uniform evaporation), all of the particles are eventually advected to the contact line, and so the final deposit is a ring deposit at the contact line, whereas when $n>1$ all of the particles are eventually advected to the centre of the droplet, and so the final deposit is at the centre of the droplet. In particular, the present work demonstrates that a singular (or even a non-zero) evaporative flux at the contact line is not an essential requirement for the formation of a ring deposit. In addition, we calculate the paths of the particles when diffusion is slower than both axial and radial advection, and show that in this regime all of the particles are captured by the descending free surface before eventually being deposited onto the substrate.

We consider the dynamics of a gravity current of viscous liquid propagating above a dense granular medium that obeys a $\mu (I)$-rheology. Initially, the pool of liquid depresses the granular layer to form levees at its edges. Next, these levees are pushed outwards by the gravity-driven slumping of the liquid, but the levees are not surmounted. In the third stage, the top of the levee is pushed out beyond the rest of the levee. This segregates the liquid into a pond trapped by the remnant of the original levees, and a slowly spreading thin film ahead of the levees. The trapped fraction of liquid depends on the extent of the early granular erosion, which in turn is controlled by the initial shape of the deposit and the yield criterion of the granular layer. The key physical ingredients that lead to such dynamics are inertia-less flow and a lower layer with a yield criterion. The latter gives rise to the all-important levees, which lead to the eventual trapping.

In conventional microfluidic devices, fluids are often confined behind solid plastic walls that restrict access and trap gas bubbles; in open microfluidics some solid walls are replaced by fluid ones (i.e. interfaces with immiscible fluids). In both cases, flows are usually driven by external pumps or gravity. An innovative open technology has been developed in which two-dimensional patterns of cell-culture medium in standard Petri dishes are confined by fluid walls made of an immiscible and bio-inert fluorocarbon (FC40). To provide refreshing media flows to cells in such circuits, an established pumping system that exploits differences in Laplace pressure across open interfaces has been applied to drive flow without using external pumps: a source drop autonomously empties through a straight conduit into the rest of the dish (the sink). Whereas conduits with solid walls have unchanging boundaries and flows within them are well understood, the challenge is to predict flows in circuits where fluid walls morph as pressures change. Numerical and semi-analytical equations enabling the prediction of changing flows are developed, and predictions validated experimentally.

Langmuir circulations (LCs) arise through the interaction between the Lagrangian drift of the surface waves and the wind-driven shear layer. Quasi-streamwise vortices (QSVs) also form in the turbulent shear layer next to a flat surface. Both vortical structures manifest themselves by inducing wind-aligned streaks on the surface. In this study, numerical simulations of a stress-driven turbulent shear layer bounded by monochromatic surface waves are conducted to reveal the vortical structures of LCs and QSVs, and their interactions. The LC structure is educed from conditional averaging guided by the signatures of predominant streaks obtained from empirical mode decomposition; the width of the averaged LC pair is found to be comparable to the most unstable wavelength of the Craik–Leibovich equation. Coherent vortical structures (CVSs) are identified using a detection criterion based on local analysis of the velocity-gradient tensor and their topological geometry; QSVs accumulated beneath the windward surface are found to dominate the distribution. Employing the variable-interval spatial average to the identified QSVs further reveals that QSVs tend to form in the edge vicinity of the surface streaks induced by the LCs. The transport budgets of streamwise enstrophy are examined to reveal the interaction. It is found that QSVs perturb the streaks resulting in a localized streamwise gradient of the spanwise velocity, that is, vertical vorticity. The vertical shear tilts the vertical vorticity, therefore enhancing streamwise enstrophy production and the formation of QSVs. The results highlight the differences in the CVSs between the Langmuir turbulence and the wall turbulence.

Polymers in a turbulent flow are stretched out by the fluctuating velocity gradient and exhibit a broad distribution of extensions $R$; the stationary probability density function (p.d.f.) of $R$ has a power-law tail with an exponent that increases with the Weissenberg number $\mathit {Wi}$, a non-dimensional measure of polymer elasticity. This study addresses the following questions. (i) What is the role of the non-Gaussian statistics of the turbulent velocity gradient on polymer stretching? (ii) How does the p.d.f. of $R$ evolve to its asymptotic stationary form? Our analysis is based on simulations of the dynamics of finitely extensible bead–spring dumbbells and chains, in the extremely dilute limit, that are transported in a homogeneous and isotropic turbulent flow, as well as in a Gaussian random flow. We show that while the turbulent flow is more effective at stretching small-$\mathit {Wi}$ stiff polymers, the Gaussian flow is more effective for high-$\mathit {Wi}$ polymers. This suggests that high-$\mathit {Wi}$ polymers (with large relaxation times) are stretched primarily by the cumulative effect of moderate strain rate events, rather than by short-lived extreme-valued strain rates. Next, we show that, beginning from a distribution of coiled polymers, the p.d.f. of $R$ exhibits two distinct regimes of evolution. At low to moderate $\mathit {Wi}$, the p.d.f. quickly develops a power-law tail with an exponent that evolves in time and approaches its stationary value exponentially. At high $\mathit {Wi}$, the rapid stretching of polymers first produces a peak in the p.d.f. near their maximum extension; a power law with a constant exponent then emerges and expands its range towards smaller $R$. The time scales of equilibration, measured as a function of $\mathit {Wi}$, point to a critical slowing down at the coil–stretch transition. Importantly, these results show no qualitative change when chains in a turbulent flow are replaced by dumbbells in a Gaussian flow, thereby supporting the use of the latter for reduced-order modelling.

The scalar dispersion of a ground-level point-source plume in a smooth-wall turbulent boundary layer is experimentally investigated using simultaneous particle image velocimetry and planar laser-induced fluorescence techniques. In the near-source region, the viscous sublayer is observed to trap dye, while in the far field, the half-width, vertical profiles and peak decay of the mean concentration and concentration variance exhibit self-similar behaviour and collapse with empirical relations. Full two-dimensional maps of the turbulent scalar fluxes show a net transport direction of upward and towards the incoming flow, with the vertical profiles collapsing well with Weibull-type exponential functions and the decay of peaks following power laws. Using the first-order gradient transport to model the turbulent scalar fluxes, maps of the anisotropic turbulent diffusivity tensor and an effective turbulent diffusivity coefficient are calculated. The streamwise and wall-normal turbulent scalar fluxes are driven dominantly by the wall-normal concentration gradient. The turbulent Schmidt number, relating the turbulent diffusivity and the turbulent (eddy) viscosity calculated using the Boussinesq hypothesis, varies with wall-normal position with values of the order of unity in the logarithmic layer.

The roll waves in open-channel flow on steep slopes can strike an obstacle with great force. We conducted two-dimensional shallow-water simulations to study the impact force of the waves against structures of various shapes and orientations. The focus is on the front runner of a wave packet developed from spatial instability. The numerical results include the stand-off distance of the bow shock wave, the front face's run-up height and the wave force on the obstacle. The strength of the impact depends on the Froude number of the undisturbed flow and the obstacle's distance from the local disturbance but not much on the form of the perturbation that initiates the instability. The wave force could reach a peak of more than an order of magnitude greater than the force on the structure without the roll waves. However, an obstacle with a sharp and pointy front can deflect the incident waves, significantly reducing the impact force.

Our understanding of the material organization of complex fluid flows has benefited recently from mathematical developments in the theory of objective coherent structures. These methods have provided a wealth of approaches that identify transport barriers in three-dimensional (3-D) turbulent flows. Specifically, theoretical advances have been incorporated into numerical algorithms that extract the most influential advective, diffusive and active barriers to transport from data sets in a frame-indifferent fashion. To date, however, there has been very limited investigation into these objectively defined transport barriers in 3-D unsteady flows with complicated spatiotemporal dynamics. Similarly, no systematic comparison of advective, diffusive and active barriers has been carried out in a 3-D flow with both thermally driven and mechanically modified structures. In our study, we utilize simulations of turbulent rotating Rayleigh–Bénard convection to uncover the interplay between advective transport barriers (Lagrangian coherent structures), material barriers to diffusive heat transport, and objective Eulerian barriers to momentum transport. For a range of (inverse) Rossby numbers, we identify each type of barrier and find intriguing relationships between momentum and heat transport that can be related to changes in the relative influence of mechanical and thermal forces. Further connections between bulk behaviours and structure-specific behaviours are also developed.

Experiments are conducted in an open-channel flow where half of the section is smooth and the other half consists of an array of cubes, which are either submerged or emergent. A shear layer featuring large-scale Kelvin–Helmholtz structures develops between the two subsections. The flows are first analysed in the framework of the double-averaging method (averaging of the flow both in time and space). Double averaging could be performed thanks to an experimental set-up (three-dimensional, two-component telecentric scanning particle image velocimetry) that allows to measure the velocity field in a large volume, including the interstices between the cubes. A momentum balance performed on the smooth subsection indicates that the loss of momentum towards the rough subsection has the same order of magnitude than the momentum loss through bed friction. This lateral momentum flux occurs nearly exclusively through turbulent shear stress, whereas secondary currents plays a minor role and dispersive shear stress is negligible. A pattern recognition technique is then applied to investigate statistically the large-scale Kelvin–Helmholtz structures that develop in the shear layer. The structures appear to be coherent over the water depth and to be strongly inclined in the vertical, the top part being ahead. The educed coherent structure is responsible by itself for the shape of the velocity profile across the shear layer and for a large part of the turbulence (up to 60 % for the turbulent shear stress). Finally, a coupling is identified between the passage of the Kelvin–Helmholtz structures and the instantaneous wake flow around the cubes at the interface.

The presence of salt in seawater strongly affects the melt rate and the shape evolution of ice, both of utmost relevance in ice–ocean interactions and thus for the climate. To get a better quantitative understanding of the physical mechanics at play in ice melting in salty water, we numerically investigate the lateral melting of an ice block in stably stratified saline water. The developing ice shape from our numerical results shows good agreement with the experiments and theory from Huppert & Turner (J. Fluid Mech., vol. 100, 1980, pp. 367–384). Furthermore, we find that the melt rate of ice depends non-monotonically on the mean ambient salinity: it first decreases for increasing salt concentration until a local minimum is attained, and then increases again. This non-monotonic behaviour of the ice melt rate is due to the competition among salinity-driven buoyancy, temperature-driven buoyancy and salinity-induced stratification. We develop a theoretical model based on the force balance which gives a prediction of the salt concentration for which the melt rate is minimal, and is consistent with our data. Our findings give insight into the interplay between phase transitions and double-diffusive convective flows.

This study examines the two-dimensional interaction of two unequal co-rotating viscous vortices in uniform background shear. Numerical simulations are performed for vortex pairs having various circulation ratios $\varLambda _0 = \varGamma _{1,0}/\varGamma _{2,0} = (\omega _{1,0}/\omega _{2,0})(a^2_{1,0}/a^2_{2,0}) \leqslant 1$, corresponding to different initial characteristic radii $a_{i,0}$ and peak vorticities $\omega _{i,0}$ of each vortex $i=1,2$, in shears of various strengths $\zeta _0 = \omega _S/\omega _{2,0}$, where $\omega _S$ is the constant vorticity of the shear. Two primary flow regimes are observed: separations ($\zeta _0 < \zeta _{sep} < 0$), in which the vortices move apart continuously, and henditions ($\zeta _0 > \zeta _{sep}$), in which the interaction results in a single vortex (where $\zeta _{sep}$ is the adverse shear strength beyond which separation occurs). Vortex motion and values of $\zeta _{sep}(\varLambda _0)$ are well-predicted by a point-vortex model for unequal vortices. In vortex-dominated henditions, shear varies the peak–peak distance $b$, and vortex deformation. The main convective interaction begins when core detrainment of one vortex is established, and proceeds similarly to the no-shear ($\zeta _0 = 0$) case: merger occurs if the second vortex also detrains, engendering mutual entrainment; otherwise straining out occurs. Detrainment requires persistence of straining of both sufficient magnitude, as indicated by relative straining above a consistent critical value, $(S/\omega )_i > (S/\omega )_{cr}$, where $S$ is the strain rate magnitude at the vorticity peak, and conducive direction. Hendition outcomes are assessed in terms of an enhancement factor $\varepsilon \equiv \varGamma _{end}/\varGamma _{2,start}$. Although $\varepsilon$ generally varies with $\zeta _0$, $(a^2_{1,0} /a^2_{2,0} )$ and $(\omega _{1,0}/\omega _{2,0})$ in a complicated manner, this variation is well-characterized by the pair's starting enstrophy ratio, $Z_2/Z_1$. Within a transition region between merger and straining out (approximately $1.65 < Z_2/Z_1 < 1.9$), shear of either sense may increase $\varepsilon$.

The use of thin liquid films has expanded beyond lubrication and coatings, and into applications in actuators and adaptive optical elements. In contrast to their predecessors, whose dynamics can be typically captured by modelling infinite or periodic films, these applications are characterized by a finite amount of liquid in an impermeable domain. The global mass conservation constraint, together with common boundary conditions (e.g. pinning), create quantitatively and qualitatively different dynamics than those of infinite films. Mathematically, this manifests itself as a non-self-adjoint problem. This work presents a combined theoretical and experimental study for this problem. We provide a time-dependent closed-form analytical solution for the linearized non-self-adjoint system that arises from these boundary conditions. We highlight that, in contrast to self-adjoint problems, here, special care should be given to deriving the adjoint problem to reconstruct the solution based on the eigenfunctions properly. We compare these solutions with those obtained for permeable and periodic boundary conditions, representing common models for self-adjoint thin-film problems. We show that, while the initial dynamics is nearly identical, the boundary conditions eventually affect the film deformation as well as its response time. To experimentally illustrate the dynamics and to validate the theoretical model, we fabricated an experimental set-up that subjects a thin liquid film to a prescribed normal force distribution through dielectrophoresis, and used high-frame-rate digital holography to measure the film deformation in real time. The experiments agree well with the model and confirm that confined films exhibit a different behaviour which could not be predicted by existing models.

Wake interaction of two rotationally oscillating cylinders in side-by-side configuration is studied experimentally at a Reynolds number of 150. Five spacing ratios, $T/D$ (ratio of centre-to-centre spacing to cylinder diameter), are considered, namely, 1.4, 1.8, 2.5, 4.0 and 7.5. Both in-phase and antiphase forcing are investigated. Oscillation amplitude is varied from ${\rm \pi} /8$ to ${\rm \pi}$, and forcing frequency, $FR$ (ratio of the oscillation frequency to the vortex-shedding frequency of a stationary cylinder) is varied from 0 to 5. The experimental investigation is done using laser-induced fluorescence, hot film anemometry and particle-image velocimetry (PIV). The interaction between the two cylinders under forcing results in new wake modes and vortex structures and a comprehensive study from the wake visualisations is conducted. Quantitative results are presented in terms of streamwise and cross-stream mean velocity profiles, centreline velocity recovery, peak velocity deficit, wake width, fluctuation intensity, circulation, vorticity contours and drag coefficient. The magnitude of streamwise velocity deficit and cross-stream velocity variation is strongly affected by the presence of second cylinder. The recirculation region behind the cylinders is found to extend further downstream with increase in the forcing. Scaling analysis is carried out to express the peak velocity deficit variation with forcing. It is observed that the relative strength of the vortices shed from inner and outer shear layers depends on the phase of oscillation. An experimental set-up for direct force measurement is designed and the drag force acting on the oscillating cylinders assembly is directly measured and the effect of forcing on the variation of ${C}_{d}$ is studied. An estimate for drag coefficient is also made from the PIV data following a detailed control volume analysis. It is observed that the set of forcing parameters that correspond to maximum and minimum drag also yield extrema in the values of circulation and fluctuation intensity.