A thin, evaporating sessile droplet with a pinned contact line containing inert particles is considered. In the limit in which the liquid flow decouples from the particle transport, we discuss the interplay between particle advection, diffusion and adsorption onto the solid substrate on which the droplet sits. We perform an asymptotic analysis in the physically relevant regime in which the Péclet number is large, i.e. ${\textit{Pe}}\gg 1$, so that advection dominates diffusion in the droplet except in a boundary layer near the contact line, and in which the ratio of the particle velocities due to substrate adsorption and diffusion is at most of order unity as ${\textit{Pe}}\rightarrow \infty$. We use the asymptotic model alongside numerical simulations to demonstrate that substrate adsorption leads to a different leading-order distribution of particle mass compared with cases with negligible substrate adsorption, with a significant reduction of the mass in the suspension – the nascent coffee ring reported in Moore et al. (J. Fluid Mech., vol. 920, 2021, A54). The redistribution leads to an extension of the validity of the dilute suspension assumption, albeit at the cost of breakdown due to the growth of the deposited layer, which are important considerations for future models that seek to accurately model the porous deposit regions.

Ionic surfactants are commonly employed to modify the rheological properties of fluids, particularly in terms of surface viscoelasticity. Concurrently, external electric fields can significantly impact the dynamics of liquid threads. A key question is how ionic surfactants affect the dynamic behaviour of threads in the presence of an electric field? To investigate this, a one-dimensional model of a liquid thread coated with surfactants within a radial electric field is established, employing the long-wave approximation. We systematically investigate the effects of dimensionless parameters associated with the surfactants, including surfactant concentration, dilatational Boussinesq number ${\textit{Bo}}_{\kappa \infty }$ and shear Boussinesq number ${\textit{Bo}}_{\mu \infty }$. The results indicate that increasing the surfactant concentration and the two Boussinesq numbers reduces both the maximum growth rate and the dominant wavenumber. In addition, both the electric field and surfactants mitigate the breakup of the liquid thread and the formation of satellite droplets. At low applied electric potentials, the surface viscosity induced by surfactants predominantly governs this suppression. Surface viscosity suppresses the formation of satellite droplets by maintaining the neck point at the centre of the liquid thread within a single disturbance wavelength. When the applied potential is high, the electric stress has two main effects: the external electric field exerts a normal pressure on the liquid thread surface, suppressing satellite droplet formation, while the internal electric field inhibits liquid drainage. Surface viscosity further stabilizes the system by suppressing flow dynamics during this process.

As a step towards realising a skin-friction drag reduction technique that scales favourably with Reynolds number, the impact of a synthetic jet on a turbulent boundary layer was explored through a study combining wind-tunnel measurements and large eddy simulations. The jet was ejected in the wall-normal direction through a rectangular slot whose spanwise dimension matched that of dominant large-scale structures in the logarithmic region to target structures of that size and smaller simultaneously. Local skin-friction reduction was observed at both $x/\delta =2$ and $x/\delta =5$ downstream of the orifice centreline, where $\delta$ is the boundary-layer thickness. At $x/\delta =2$, the skin-friction reduction was observed to be due to the synthetic-jet velocity deficit intersecting the wall. At $x/\delta =5$, evidence from the simulations and wind-tunnel measurements suggests that a weakening of wall-coherent velocity scales is primarily responsible for the skin-friction reduction. Local skin-friction reduction which scales favourably with Reynolds number may be achievable with the synthetic jet employed in this study. However, there are many technical hurdles to overcome to achieve net skin-friction drag reduction over the entire region of influence. For instance, regions of skin-friction increase were observed close to the orifice ($x/\delta \lt 2$) and downstream of the orifice edge due to the induced motion of synthetic-jet vortical structures. Additionally, a recirculation region was seen to form during expulsion, which has implications for pressure drag on non-planar surfaces.

The triadic interactions and nonlinear energy transfer are investigated in a subsonic turbulent jet at $Re = 450\,000$. The primary focus is on the role of these interactions in the formation and attenuation of streaky structures. To this end, we employ bispectral mode decomposition, a technique that extracts coherent structures associated with dominant triadic interactions. A strong triadic correlation is identified between Kelvin–Helmholtz (KH) wavepackets and streaks: interactions between counter-rotating KH waves generates streamwise vortices, which subsequently give rise to streaks through the lift-up mechanism. The most energetic streaks occur at azimuthal wavenumber $m = 2$, with the dominant contributing triad being $[m_1, m_2, m_3] = [1, 1, 2]$. The spectral energy budget reveals that the net effect of nonlinear triadic interactions is an energy loss from the streaks. As these streaks convect downstream, they engage in further nonlinear interactions with other frequencies, which drain their energy and ultimately lead to their attenuation. Further analysis identifies the dominant scales and direction of energy transfer across different spatial regions of the jet. While the turbulent jet exhibits a forward energy cascade in a global sense, the direction of energy transfer varies locally: in the shear layer near the nozzle exit, triadic interactions among smaller scales dominate, resulting in an inverse energy cascade, whereas farther downstream, beyond the end of the potential core, interactions among larger scales prevail, leading to a forward cascade.

Secondary flows induced by spanwise heterogeneous surface roughness play a crucial role in determining engineering-relevant metrics such as surface drag, convective heat transfer and the transport of airborne scalars. While much of the existing literature has focused on idealized configurations with regularly spaced roughness elements, real-world surfaces often feature irregularities, clustering and topographic complexity for which the secondary flow response remains poorly understood. Motivated by this gap, we investigate multicolumn roughness configurations that serve as a regularized analogue of roughness clustering. Using large-eddy simulations, we systematically examine secondary flows across a controlled set of configurations in which cluster density and local arrangement are varied in an idealized manner, and observe that these variations give rise to distinct secondary flow polarities. Through a focused parameter study, we identify the spanwise gap between the edge-most roughness elements of adjacent columns, normalized by the channel half-height ($s_a/H$), as a key geometric factor governing this polarity. In addition to analysing the time-averaged structure, we investigate how variations in polarity affect the instantaneous dynamics of secondary flows. Here, we find that the regions of high- and low-momentum fluid created by the secondary flows alternate in a chaotic, non-periodic manner over time. Further analysis of the vertical velocity signal shows that variability in vertical momentum transport is a persistent and intrinsic feature of secondary flow dynamics. Taken together, these findings provide a comprehensive picture of how the geometric arrangement of roughness elements governs both the mean structure and temporal behaviour of secondary flows.

A fully resolved numerical study was performed to investigate interfacial heat and mass transfer enhanced by the fully developed Rayleigh–Bénard–Marangoni instability in a relatively deep domain. The instability was triggered by evaporative cooling modelled by a constant surface heat flux. The latter allowed for temperature-induced variations in surface tension giving rise to Marangoni forces reinforcing the Rayleigh instability. Simulations were performed at a fixed Rayleigh number (${\textit{Ra}}_h$) and a variety of Marangoni numbers (${\textit{Ma}}_h$). In each simulation, scalar transport equations for heat and mass concentration at various Schmidt numbers (${\textit{Sc}}=16{-}200$) were solved simultaneously. Due to the fixed (warm) temperature prescribed at the bottom of the computational domain, large buoyant plumes emerged quasi-periodically both at the top and bottom. With increasing Marangoni number a decrease in the average convection cell size at the surface was observed, with a simultaneous improvement in near-surface mixing. The presence of high aspect ratio rectangular convection cell footprints was found to be characteristic for Marangoni-dominated flows. Due to the promotion of interfacial mass transfer by Marangoni forces, the power in the scaling of the mass transfer velocity, $K_{\!L}\!\propto\! \textit{Sc}^{-n}$, was found to decrease from $n=0.50$ at ${\textit{Ma}}_h=0$ to $\approx 0.438$ at ${\textit{Ma}}_h=13.21\times 10^5$. Finally, the existence of a buoyancy-dominated and a Marangoni-dominated regime was investigated in the context of the interfacial heat and mass transfer scaling as a function of ${\textit{Ma}}_h+\varepsilon {\textit{Ra}}_h$, where $\varepsilon$ is a small number determined empirically.

Both experiments and direct numerical simulation (DNS) of hypersonic flow over a compression ramp show streamwise aligned streaks/vortices near the corner as the ramp angle is increased. The origin of this three-dimensional disturbance growth is not definitively known in the existing literature, but is typically connected to flow deceleration, centrifugal (Görtler) and/or baroclinic effects. In this work we consider the hypersonic problem with moderate wall cooling in the high Reynolds/Mach number, weak interaction limit. In the lower deck of the corresponding asymptotic triple-deck description we pose the linearised, three-dimensional, Görtler stability equations. This formulation allows computation of both receptivity and biglobal stability problems for linear spanwise-periodic disturbances with a spanwise wavelength of the same order as the lower-deck depth. In this framework the dominant response near the ramp surface is of constant density and temperature (at leading order) ruling out baroclinic mechanisms. Nevertheless, we show that there remains strong energy growth of upstream spanwise-varying perturbations and ultimately a bifurcation from two-dimensional to three-dimensional ramp flow. The unstable eigenmodes are localised to the separation region. The bifurcation points are obtained over a range of ramp angle, wall-cooling parameter and disturbance wavelength. Consistent with DNS results, the three-dimensional perturbations in this asymptotic formulation are streamwise aligned streaks/vortices, displaced above the separation region. In addition, the growth of upstream disturbances peaks near to the reattachment point, whilst the streaks persist beyond it, decaying relatively slowly downstream along the deflected ramp.

Sea surface films significantly influence air–sea interaction. While their damping effect on gravity–capillary waves is well recognised, the detailed mechanisms by which surface films alter small-scale wave dynamics – particularly energy dissipation and near-surface flow patterns – remain insufficiently understood. This paper presents experimental observations focusing on small-scale wave profiles and surface-flow dynamics in the presence of surfactants, providing direct experimental evidence of underlying mechanisms such as Marangoni effects. The experiments demonstrate enhanced energy dissipation and significant alterations in near-surface flow caused by surfactants, including the transformation of typical circular motion into elliptical-like trajectories and the emergence of reverse surface drift.

Spectral analysis of the transport process of turbulence kinetic energy (TKE) in a channel roughened with spanwise-aligned circular-arc ribs is conducted based on direct numerical simulations (DNS). Test cases of varying pitch-to-height ratios ($P/H=3.0$, 5.0 and 7.5) and bulk Reynolds numbers (${\textit{Re}}_b=5600$ and 14 600) are compared. It is observed that the characteristic spanwise wavelength of the energy-containing eddies in the internal shear layer (ISL) increases as the value of $P/H$ increases, but decreases as the Reynolds number increases. In the ISL, the energy transport processes are dominated by turbulent production as the lead source term, but by turbulent diffusion and dissipation as the lead sink terms. It is found that regions with high production and dissipation rates of TKE in the ISL are associated with moderate and small wavelengths, respectively. The TKE production for sustaining moderate- and large-scale motions enhances gradually with an increasing value of $P/H$, while that for sustaining small-scale motions augments as the Reynolds number increases. It is interesting to observe that the interscale-transport term plays a critical role in draining TKE at moderate wavelengths as a sink and carries the drained TKE to small-scale eddies as a source. It is discovered that a higher pitch-to-height ratio leads to shortening of the characteristic spanwise wavelength of the dissipation process but prolongation of those of the production, interscale-transport and turbulent-diffusion processes in the ISL. By contrast, a higher Reynolds number results in reductions in the characteristic spanwise wavelengths of all spectral transport terms.

The emergence of large-scale spatial modulations of turbulent channel flow, as the Reynolds number is decreased, is addressed numerically using the framework of linear stability analysis. Such modulations are known as the precursors of laminar–turbulent patterns found near the onset of relaminarisation. A synthetic two-dimensional base flow is constructed by adding finite-amplitude streaks to the turbulent mean flow. The streak mode is chosen as the leading resolvent mode from linear response theory. In addition, turbulent fluctuations can be taken into account or not by using a simple Cess eddy viscosity model. The linear stability of the base flow is considered by searching for unstable eigenmodes with wavelengths larger than the base flow streaks. As the streak amplitude is increased in the presence of the turbulent closure, the base flow loses its stability to a large-scale modulation below a critical Reynolds-number value. The structure of the corresponding eigenmode, its critical Reynolds number, its critical angle and its wavelengths are all fully consistent with the onset of turbulent modulations from the literature. The existence of a threshold value of the Reynolds number is directly related to the presence of an eddy viscosity, and is justified using an energy budget. The values of the critical streak amplitudes are discussed in relation with those relevant to turbulent flows.

We propose a novel multiple-scale spatial marching method for flows with slow streamwise variation. The key idea is to couple the boundary region equations, which govern large-scale flow evolution, with local exact coherent structures that capture the small-scale dynamics. This framework is consistent with high-Reynolds-number asymptotic theory and offers a promising approach to constructing time-periodic finite-amplitude solutions in a broad class of spatially developing shear flows. As a first application, we consider a non-uniformly curved channel flow, assuming that a finite-amplitude travelling-wave solution of plane Poiseuille flow is sustained at the inlet. The method allows for the estimation of momentum transport and highlights the impact of the inlet condition on both the transport properties and the overall flow structure. We then consider a case with gradually decreasing curvature, starting with Dean vortices at the inlet. In this setting, small external oscillatory disturbances can give rise to subcritical self-sustained states that persist even after the curvature vanishes.

In the paper, we consider a two-dimensional free-surface flow past a single point vortex in fluid of infinite depth. The flow moves from left to right with uniform speed $c$ far upstream and is subject to the downward acceleration $g$ of gravity. A point vortex of circulation $\varGamma$ is located at depth $H$. The positive direction of circulation is counterclockwise. The flow is characterised by two dimensionless parameters which are the dimensionless vortex circulation $\gamma =\varGamma /(\textit{cH}\,)$ and the Froude number $ \textit{Fr}=c/\sqrt {gH}$. The goal of the paper is to find the solutions of the solitary wave type with one or several crests on the free surface. These solutions are waveless far downstream and have a vertical line of symmetry. We have established that for a fixed Froude number $ \textit{Fr}\le 0.8$, there exists a finite set of positive $\gamma$ for which the solutions of the solitary wave type occur.

Granular flow down an inclined plane is ubiquitous in geophysical and industrial applications. On rough inclines, the flow exhibits Bagnold’s velocity profile and follows the so-called $\mu (I)$ local rheology. On insufficiently rough or smooth inclines, however, velocity slip occurs at the bottom and a basal layer with strong agitation emerges below the bulk, which is not predicted by the local rheology. Here, we use discrete element method simulations to study detailed dynamics of the basal layer in granular flows down both smooth and rough inclines. We control the roughness via a dimensionless parameter, $R_a$, varied systematically from 0 (flat, frictional plane) to near 1 (very rough plane). Three flow regimes are identified: a slip regime ($R_a \lesssim 0.45$) where a dilated basal layer appears, a no-slip regime ($R_a \gtrsim 0.6$) and an intermediate transition regime. In the slip regime the kinematics profiles (velocity, shear rate and granular temperature) of the basal layer strongly deviate from Bagnold’s profiles. General basal slip laws are developed that express the slip velocity as a function of the local shear rate (or granular temperature), base roughness and slope angle. Moreover, the basal layer thickness is insensitive to flow conditions but depends somewhat on the interparticle coefficient of restitution. Finally, we show that the rheological properties of the basal layer do not follow the $\mu (I)$ rheology, but are captured by Bagnold’s stress scaling and an extended kinetic theory for granular flows. Our findings can help develop more predictive granular flow models in the future.