Wall turbulence consists of various sizes of vortical structures that induce flow circulation around a wide range of closed Eulerian loops. Here we investigate the multiscale properties of circulation around such loops in statistically homogeneous planes parallel to the wall. Using a high-resolution direct numerical simulation database of turbulent channels at Reynolds numbers of $Re_\tau =180$, 550, 1000 and 5200, circulation statistics are obtained in planes at different wall-normal heights. Intermittency of circulation in the planes of the outer flow ($y^+ \gtrsim 0.1Re_\tau$) takes the form of universal bifractality as in homogeneous and isotropic turbulence. The bifractal character simplifies to space-filling character close to the wall, with scaling exponents that are linear in the moment order, and lower than those given by the Kolmogorov paradigm. The probability density functions of circulation are long-tailed in the outer bifractal region, with evidence showing their invariance with respect to the loop aspect ratio, while those in the inner region are closely Gaussian. The unifractality near the wall implies that the circulation there is not intermittent in character.

The aspect ratio effect on side and basal melting in fresh water is systematically investigated across a range of Rayleigh numbers and ambient temperatures using direct numerical simulations. The side mean melt rate follows a ${Ra}^{1/4}\,\gamma ^{-3/8}$ scaling relation in the side-melting dominant regime, where ${Ra}$ is the Rayleigh number, and $\gamma$ is the width-to-height aspect ratio of the ice block. In the basal-melting dominant regime, the basal mean melt rate follows a ${Ra}^{1/4}\gamma ^{3/8}$ scaling relation at low Rayleigh numbers, but transitions to a ${Ra}^{1/3}\gamma ^{1/2}$ scaling relation at higher Rayleigh numbers. This scaling transition is attributed to the formation of a bottom cavity resulting from flow separation at high Rayleigh numbers. The overall mean melt rate exhibits a non-monotonic dependence on the aspect ratio, driven by the competition between side and basal melting. The proposed theoretical model successfully captures the observed non-monotonic behaviour, and accurately predicts the overall mean melt rate over the considered range of Rayleigh numbers and ambient temperatures, especially in the side- and basal-melting dominant regimes. More specifically, the side, basal and overall mean melt rates follow a linear ${St}$ scaling relation for ambient temperatures $T_{w}\geqslant 15^{\,\circ }\textrm {C}$, with ${St}$ being the Stefan number (the ratio between sensible heat and latent heat), but deviations from this scaling relation and a non-monotonic dependence on the ambient temperature are observed at lower ambient temperatures, which can be attributed to the density anomaly effect.

This paper investigates the behaviour of turbulence production in adverse pressure gradient (APG) turbulent boundary layers (TBLs), including the range of pressure gradients from zero-pressure-gradient (ZPG) to separation, moderate and high Reynolds numbers, and equilibrium and non-equilibrium flows. The main focus is on predicting the values and positions of turbulence production peaks. Based on the unique ability of turbulence production to describe energy exchange, the idea that the ratios of the mean flow length scales to the turbulence length scales are locally smallest near peaks is proposed. Thereby, the ratios of length scales are defined for the inner and outer regions, respectively, as well as the ratios of time scales for further consideration of local information. The ratios in the inner region are found to reach the same constant value in different APG TBLs. Like turbulence production in the ZPG TBL, turbulence production in APG TBLs is shown to have a certain invariance of the inner peak. The value and position of the inner peak can also be predicted quantitatively. In contrast, the ratios in the outer region cannot be determined with unique coefficients, which accounts for the different self-similarity properties of the inner and outer regions. The outer time scale ratios establish a link between mean flow and turbulence, thus participating in the discussion on half-power laws. The present results support the existence of a half-power-law region that is not immediately adjacent to the overlapping region.

This article delves into the dynamics of inviscid annular supersonic jets, akin to those exiting converging–diverging nozzles in over-expanded regimes. It focuses on the first azimuthal Fourier mode of flow fluctuations and examines their behaviour with varying mixing layer parameters and expansion regimes. The study reveals that two unstable Kelvin–Helmholtz waves exist in all cases, with the outer-layer wave being more unstable due to differences in the velocity gradient. The inner-layer wave is more sensitive to changes in base flow and extends beyond the jet, potentially contributing to nozzle resonances. The article also investigates upstream propagating guided-jet modes, which are found to be robust and not highly sensitive to changes in base flow, which makes them essential for understanding jet dynamics. A simplified model is used to obtain ideal base flows but with realistic shape in order to study the effects of varying nozzle pressure ratios on the dynamics of the waves supported by the jet.

Droplet coalescence is an essential multiphase flow process in nature and industry. For the inviscid coalescence of two spherical droplets, our experiment shows that the classical 1/2 power-law scaling for equal-size droplets still holds for the unequal-size situation of small size ratios, but it diverges as the size ratio increases. Employing an energy balance analysis, we develop the first theory for asymmetric droplet coalescence, yielding a solution that collapses all experimental data of different size ratios. This confirms the physical relevance of the new set of length and time scales given by the theory. The functionality of the solution reveals an exponential dependence of the bridge’s radial growth on time, implying a scaling-free nature. Nevertheless, the small-time asymptote of the model is able to recover the classical power-law scaling, so that the actual bridge evolution still follows the scaling law asymptotically in a wide parameter space. Further analysis suggests that the scaling-free evolution behaviour emerges only at late coalescence time and large size ratios.

An experimental study is conducted to compare droplet generation in a deep-water plunging breaker in filtered tap water and in the presence of low and high bulk concentrations of the soluble surfactant Triton X-100. The breakers are generated by a programmable wave maker that is set with a single motion profile that produces a highly repeatable dispersively focused two-dimensional (2-D) wave packet with a central wavelength of $\lambda _0=1.18\,\rm m$. The droplets are measured with an in-line cinematic holographic system. It is found that the presence of surfactants significantly modifies the overall droplet number and the distributions of droplet diameter and velocity components produced by the four main droplet producing mechanisms of the breaker as identified by Erinin et al. ( J. Fluid Mech., vol. 967, 2023, p. A36). These modifications are due to both surfactant-induced changes in the flow structures that generate droplets and changes in the details of droplet production mechanisms in each flow structure.

Thermo-responsive hydrogels are smart materials that rapidly switch between hydrophilic (swollen) and hydrophobic (shrunken) states when heated past a threshold temperature, resulting in order-of-magnitude changes in gel volume. Modelling the dynamics of this switch is notoriously difficult and typically involves fitting a large number of microscopic material parameters to experimental data. In this paper, we present and validate an intuitive, macroscopic description of responsive gel dynamics and use it to explore the shrinking, swelling and pumping of responsive hydrogel displacement pumps for microfluidic devices. We finish with a discussion on how such tubular structures may be used to speed up the response times of larger hydrogel smart actuators and unlock new possibilities for dynamic shape change.

Bubble bursting and subsequent collapse of the open cavity at free surfaces of contaminated liquids can generate aerosol droplets, facilitating pathogen transport. After film rupture, capillary waves focus at the cavity base, potentially generating fast Worthington jets that are responsible for ejecting the droplets away from the source. While extensively studied for Newtonian fluids, the influence of non-Newtonian rheology on this process remains poorly understood. Here, we employ direct numerical simulations to investigate the bubble cavity collapse in viscoelastic media, such as polymeric liquids. We find that the jet and drop formations are dictated by two dimensionless parameters: the elastocapillary number $Ec$ (the ratio of the elastic modulus and the Laplace pressure) and the Deborah number $De$ (the ratio of the relaxation time and the inertio-capillary time scale). We show that, for low values of $Ec$ and $De$, the viscoelastic liquid adopts a Newtonian-like behaviour, where the dynamics is governed by the solvent Ohnesorge number $Oh_s$ (the ratio of visco-capillary and inertio-capillary time scales). In contrast, for large values $Ec$ and $De$, the enhanced elastic stresses completely suppress the formation of the jet. For some cases with intermediate values of $Ec$ and $De$, smaller droplets are produced compared with Newtonian fluids, potentially enhancing aerosol dispersal. By mapping the phase space spanned by $Ec$, $De$ and $Oh_s$, we reveal three distinct flow regimes: (i) jets forming droplets, (ii) jets without droplet formation and (iii) absence of jet formation. Our results elucidate the mechanisms underlying aerosol suppression versus fine spray formation in polymeric liquids, with implications for pathogen transmission and industrial processes involving viscoelastic fluids.

The Cahn–Hilliard–Navier–Stokes (CHNS) partial differential equations (PDEs) provide a powerful framework for the study of the statistical mechanics and fluid dynamics of multiphase fluids. We provide an introduction to the equilibrium and non-equilibrium statistical mechanics of systems in which coexisting phases, distinguished from each other by scalar order parameters, are separated by an interface. We then introduce the coupled CHNS PDEs for two immiscible fluids and generalisations for (i) coexisting phases with different viscosities, (ii) CHNS with gravity, (iii) three-component fluids and (iv) the CHNS for active fluids. We discuss mathematical issues of the regularity of solutions of the CHNS PDEs. Finally we provide a survey of the rich variety of results that have been obtained by numerical studies of CHNS-type PDEs for diverse systems, including bubbles in turbulent flows, antibubbles, droplet and liquid-lens mergers, turbulence in the active-CHNS model and its generalisation that can lead to a self-propelled droplet.

This investigation examines the dynamic response of an accelerating turbulent pipe flow using direct numerical simulation data sets. A low/high-pass Fourier filter is used to investigate the contribution and time dependence of the large-scale motions (LSM) and the small-scale motions (SSM) into the transient Reynolds shear stress. Additionally, it analyses how the LSM and SSM influence the mean wall shear stress using the Fukagata–Iwamoto–Kasagi identity. The results reveal that turbulence is frozen during the early flow excursion. During the pretransition stage, energy growth of the LSM and a subtle decay in the SSM is observed, suggesting a laminarescent trend of SSM. The transition period exhibits rapid energy growth in the SSM energy spectrum at the near-wall region, implying a shift in the dominant contribution from LSM to SSM to the frictional drag. The core-relaxation stage shows a quasisteady behaviour in large- and small-scale turbulence at the near-wall region and progressive growth of small- and large-scale turbulence within the wake region. The wall-normal gradient of the Reynolds shear stress premultiplied energy cospectra was analysed to understand how LSM and SSM influence the mean momentum balance across the different transient stages. A relevant observation is the creation of a momentum sink produced at the buffer region in large- and very large-scale (VLSM) wavelengths during the pretransition. This sink region annihilates a momentum source located in the VLSM spectrum and at the onset of the logarithmic region of the net-force spectra. This region is a source term in steady wall-bounded turbulence.

This paper discusses the propagation of coastal currents generated by a river outflow using a 1 ${1}/{2}$-layer, quasigeostrophic model, following Johnson et al. (2017) (JSM17). The model incorporates two key physical processes: Kelvin-wave-generated flow and vortical advection along the coast. We extend JSM17 by deriving a fully nonlinear, long-wave, dispersive equation governing the evolution of the coastal current width. Numerical solutions show that, at large times, the flow behaviour divides naturally into three regimes: a steady outflow region, intermediate regions consisting of constant-width steady currents and unsteady propagating fronts leading the current. The widths of the steady currents depend strongly on dispersion when the constant outflow potential-vorticity anomaly is negative. Simulations using contour dynamics show that the dispersive equation captures the full quasigeostrophic behaviour more closely than JSM17 and give accurate bounds on the widths of the steady currents.

We study the evaporation dynamics of non-thin non-spherical-cap (i.e. wavy) droplets. These droplets exhibit surface curvature that varies periodically with the polar angle, which profoundly influences their evaporation flux, internal flow dynamics, and the resultant deposition patterns upon complete evaporation. The droplet is considered quasi-static throughout its entire lifetime. The asymptotic expansions of the evaporation flux in the diffusion-limited model, and the induced internal inviscid flow of the droplets, are derived through asymptotic analysis. Under the assumption of small deformation amplitudes, the accuracies of these two expansions are validated numerically. Expanding upon these asymptotic results, we also investigate the surface density profile of the droplet deposition after it dries up. The results indicate that the freely moving contact line of the droplet leads to the deposited stain exhibiting a mountain-like morphology. The internal inviscid flow along with the non-spherical-cap shape eliminates the divergence of the deposited surface density profile at droplet’s centre. This work provides a theoretical basis for geometrically controlled sessile droplet evaporation, which may have practical applications in industry.