A horizontal cylinder with a concave cross-section partially submerged in a liquid at a given position may permit multiple menisci around itself. The number and stabilities of the menisci are analysed, and how the menisci change during the processes of gradually hoisting and lowering the cylinder is explained by bifurcation theory. The restoring force on the concave cylinder and the rebounding potential energy (defined as the work done by the restoring force during the whole hoisting process to represent the potential rebounding capacity of a cylinder on water) are also investigated. The results show that, when the radius of the concave arc is smaller than the critical value, the concave cylinder at a given position permits multiple possible menisci. The equilibria form fold bifurcations with the position of the cylinder as the bifurcation parameter, and two successive fold bifurcations can form a one-fold hysteresis loop. The force–distance curve representing the relation between the restoring force and the position of the cylinder also has corresponding hysteresis loops, where the restoring force will jump (i.e. change discontinuously) at the bifurcation points. In contrast to a convex cylinder, a concave cylinder can have different values of the restoring force at the same height because of multiple menisci, and the values depend on whether it is hoisted or lowered. Under the condition of a fixed cross-sectional area, the optimal cross-sectional shape is determined when the maximum rebounding potential energy is reached, and it is close to the shape with the critical concave arc angle for the existence of multiple possible menisci. The cross-sections with concave parts are preferable to circular, laterally planed and corner-concave cross-sections. This paper provides an effective method of enhancing the restoring force and potential rebounding height of a robotic water strider insect or particles on the surface of water.

Using direct numerical simulations, two-dimensional tilted Rayleigh–Bénard convection (RBC) is studied in both smooth and roughness-facilitated convection cells of double aspect ratio ($\varGamma =2$) for air as a working fluid. We investigate the effect of inclination angle ($0^{\circ } \leq \phi \leq 90^{\circ }$) on heat flux ($Nu$), Reynolds number ($Re$) and flow structures. In a Rayleigh number range $10^{6}\leq Ra\leq 10^{9}$, we address the $Ra$ dependence of $Nu(\phi )$ trend. In the smooth case, while greater tilt results in highest heat flux below $Ra=10^{8}$, $Nu$ drops with $\phi$ monotonically above it (RBC transports heat most efficiently), which explains the different $Nu(\phi )$ trend observed in the previous studies due to $Ra$ dependence (Guo et al., J. Fluid Mech., vol. 762, 2015, pp. 273–287; Shishkina & Horn, J. Fluid Mech., vol. 790, 2016, R3; Khalilov et al., Phys. Rev. Fluids, vol. 3, 2018, 043503). For the smooth case, we identify the control parameters ($\phi =75^{\circ }$ and $Ra=10^{7}$) that yield maximum heat flux (an increment of $18\,\%$ with respect to the level case). On the other hand, among the three roughness set-ups used in the present study, the tallest roughness configuration yields the maximum increment in heat flux ($25\,\%$) in vertical convection ($\phi =90^{\circ }$) at $Ra=10^{6}$. With increase in $Ra$, $Re$ changes with $\phi$ marginally in the smooth case, whereas it shows notable changes in its roughness counterpart. We find that the weakening of thermal stratification is related directly to the height of roughness peaks. While $Ra$ delays the onset of thermal stratification (in terms of inclination angle) in the smooth case, an increase in roughness height plays the same role in roughness-facilitated convection cells.

Characterising interfacial and unsaturated flows in heterogeneous porous layers is of both fundamental and practical interest. Under the assumption of vertical gravitational–capillary equilibrium, we present a theoretical model to describe one-dimensional flows in a porous layer with vertical variations in average pore size, porosity, intrinsic permeability and capillary pressure jump between invading and displaced fluids. The model leads to asymptotic solutions for the saturation distribution and outer envelope of the invading fluid, and for the background pressure drop across the porous layer. Eight dimensionless parameters are recognised after appropriate non-dimensionalisation of the governing equations, the influence of which is demonstrated through a series of example calculations. In particular, four asymptotic regimes are identified, representing unconfined sharp-interface flows, confined sharp-interface flows, unconfined unsaturated flows and confined unsaturated flows. Finally, in the context of flow upscaling, analytical solutions are derived for the effective relative permeability curves on the basis of exact solutions of the saturation field and interface shape, shedding light on the subtle influence of competition between injection/pumping and gravitational forces, wetting and capillary effects, viscosity contrast between the invading and displaced fluids and vertical heterogeneity of the porous layer.

We study the collisions in a gaseous medium of a dilute bidisperse suspension of non-Brownian spherical particles sedimenting along the flow axis of a simple shear flow. Continuum lubrication forces prevent particles from coming into contact in a finite time, thus collisions can occur only due to attractive interactions such as the van der Waals force. However, in a low-pressure medium, the lubrication forces are weaker than their continuum counterparts and allow particle pairs to collide, even without any attractive forces. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the interacting spheres, captures the significance of non-continuum interactions. We use uniformly valid hydrodynamic mobility functions, reflecting non-continuum lubrication at small separations and full continuum hydrodynamic interactions at moderate to larger separations. Due to the nature of the pair trajectory topology, the collision efficiency vanishes at a critical Knudsen number when simple shear flow alone drives the dynamics. Thus we perform collision calculations where particles experience the combined effects of van der Waals attraction and non-continuum hydrodynamics; van der Waals interactions enable collisions below the critical Knudsen number. Next, we calculate the collision efficiency for coupled differential sedimentation and simple shear driven motion in the presence of van der Waals interaction and non-continuum hydrodynamics. Finally, we explore the role of small particle inertia on relative trajectories and collision efficiencies in a non-continuum gas subject to a simple shear flow, ignoring the van der Waals force and gravity.

Odd viscosity is associated with a number of striking physical phenomena: a disk rotating in a viscous liquid experiences a normal compressive or tensile stress depending on its sense of rotation; chiral active matter interfacial deformations level off in a manner akin to surface tension; and unstable thermocapillary films become stabilized. In this article we show that odd viscosity decreases the size and intensity of Moffatt vortices in a corner formed by a solid substrate and a free interface. The effect is present because odd viscosity creates a normal-stress-induced shear stress at the liquid–gas interface. The new shear stress is out of phase and oppositely directed with respect to the interfacial liquid velocity. It squeezes the vortices and damps their rotation. As a consequence, Moffatt vortices can now be observed with greater facility in an experiment. Odd viscosity also increases the range of corner angles over which Moffatt vortices can be observed, and, in general, it shows a tendency to moderate singular behaviour.

The three-dimensional instability of the flow past a 5 : 1 rectangular cylinder is investigated via Floquet analysis and direct numerical simulations. A quasi-subharmonic (QS) unstable mode is detected, marking an important difference with the flow past bodies with lower aspect ratio and/or with smooth leading edge. The QS mode becomes unstable at Reynolds number (based on the cylinder thickness and free-stream velocity) $Re \approx 480$; its spanwise wavelength is approximately three times the cylinder thickness. The structural sensitivity locates the wavemaker region over the longitudinal sides of the cylinder, indicating that the instability is triggered by the mutual inviscid interaction of vortices generated by the leading edge shear layer.

Enstrophy production and flow topology are numerically investigated for statistically stationary compressible isotropic turbulence in vibrational non-equilibrium with a large-scale thermal forcing. The net enstrophy production term is decomposed into solenoidal, dilatational and isotropic dilatational terms based on the Helmholtz decomposition. From the full flow field perspective, the net enstrophy production mainly stems from the solenoidal term. For the dilatational and isotropic dilatational terms, although their local magnitudes can be considerable, the positive values in the compression region and the negative values in the expansion region cancel out on average. For the solenoidal component of the deviatoric strain-rate tensor, the statistical properties of its eigenvalues and alignments between vorticity and its eigenvectors are nearly independent of the local dilatation and vibrational relaxation. The solenoidal components of enstrophy production along three eigendirections are thus mainly affected by the vorticity. For the dilatational component of deviatoric strain-rate tensor, the statistical properties of its eigenvalues and alignments between vorticity and its eigenvectors closely relate to the local dilatation and vibrational relaxation. The dilatational components of enstrophy production along three eigendirections are therefore affected by the vorticity, eigenvalues and alignments between the vorticity and eigenvectors. The topological classification proposed by Chong et al. (Phys. Fluids, vol. 2, issue 5, 1990, pp. 765–777) is employed to decompose the flow field into various flow topologies. In the strong compression and strong expansion regions, the relaxation effects on the volume fractions of flow topologies and their relative contributions to the local enstrophy production are significant.

Turbulent mixed convection in channel flows with heterogeneous surfaces is studied using direct numerical simulations. The relative importance of buoyancy and shear effects, characterised by the bulk Richardson number $Ri_b$, is varied in order to cover the flow regimes of forced, mixed and natural convection, which are associated with different large-scale flow organisations. The heterogeneous surface consists of streamwise-aligned ridges, which are known to induce secondary motion in the case of forced convection. The large-scale streamwise rolls emerging under smooth-wall mixed convection conditions are significantly affected by the heterogeneous surfaces and their appearance is considerably reduced for dense ridge spacings. It is found that the formation of these rolls requires larger buoyancy forces than over smooth walls due to the additional drag induced by the ridges. Therefore, the transition from forced convection structures to rolls is delayed towards larger $Ri_b$ for spanwise heterogeneous surfaces. The influence of the heterogeneous surface on the flow organisation of mixed convection is particularly pronounced in the roll-to-cell transition range, where ridges favour the transition to convective cells at significantly lower $Ri_b$. In addition, the convective cells are observed to align perpendicular to the ridges with decreasing ridge spacing. We attribute this reorganisation to the fact that flow parallel to the ridges experience less drag than flow across the ridges, which is energetically more beneficial. Furthermore, we find that streamwise rolls exhibit a very slow dynamics for $Ri_b=1$ and $Ri_b=3.2$ when the ridge spacing is of the order of the rolls’ width. For these cases the up- and downdrafts of the rolls move slowly across the entire channel instead of being fixed in space, as observed for the smooth-wall cases.

We reveal the effects of sweep on the wake dynamics around NACA 0015 wings at high angles of attack using direct numerical simulations and resolvent analysis. The influence of sweep on the wake dynamics is considered for sweep angles from $0^\circ$ to $45^\circ$ and angles of attack from $16^\circ$ to $30^\circ$ for a spanwise periodic wing at a chord-based Reynolds number of $400$ and a Mach number of $0.1$. Wing sweep affects the wake dynamics, especially in terms of stability and spanwise fluctuations with implications on the development of three-dimensional (3-D) wakes. We observe that wing sweep attenuates spanwise fluctuations. Even as the sweep angle influences the wake, force and pressure coefficients can be collapsed for low angles of attack when examined in wall-normal and wingspan-normal independent flow components. Some small deviations at high sweep and incidence angles are attributed to vortical wake structures that impose secondary aerodynamic loads, revealed through the force element analysis. Furthermore, we conduct global resolvent analysis to uncover oblique modes with high disturbance amplification. The resolvent analysis also reveals the presence of wavemakers in the shear-dominated region associated with the emergence of 3-D wakes at high angles of attack. For flows at high sweep angles, the optimal convection speed of the response modes is shown to be faster than the optimal wavemakers speed suggesting a mechanism for the attenuation of perturbations. The present findings serve as a fundamental stepping stone to understanding separated flows at higher Reynolds numbers.

Using numerical simulations, we investigate the near-field temporal variability of axisymmetric helium plumes as a function of inlet-based Richardson (${Ri}_0$) and Reynolds (${Re}_0$) numbers. Previous studies have shown that ${Ri}_0$ plays a leading-order role in determining the frequency at which large-scale vortices are produced (commonly called the ‘puffing’ frequency). By contrast, ${Re}_0$ dictates the strength of localized gradients, which are important during the transition from laminar to turbulent flow. The simulations presented here span a range of ${Ri}_0$ and ${Re}_0$, and use adaptive mesh refinement to achieve high spatial resolutions. We find that as ${Re}_0$ increases for a given ${Ri}_0$, the puffing motion undergoes a transition at a critical ${Re}_0$, marking the onset of chaotic dynamics. Moreover, the critical ${Re}_0$ decreases as ${Ri}_0$ increases. When the puffing instability is non-chaotic, time series of different variables are well-correlated, exhibiting only modest changes in the dynamics (including period doubling and flapping). Once the flow becomes chaotic, denser ambient fluid penetrates the core of the plume, similar to penetrating ‘spikes’ formed by Rayleigh–Taylor instabilities, leading to only moderately correlated flow variables. These changes result in a non-trivial dependence of the puffing frequency on ${Re}_0$. Specifically, at sufficiently low ${Re}_0$, the puffing frequency falls below the prediction from Wimer et al. (J. Fluid Mech., vol. 895, 2020). As ${Re}_0$ increases beyond the critical ${Re}_0$, the puffing frequency increases and then drops back down to the predicted scaling. The dependence of the puffing frequency on ${Re}_0$ provides a possible explanation for previously observed changes in the scaling of the puffing frequency for high ${Ri}_0$.

Hydroelastic experiments have been carried out in a free surface water channel to study the influence of the cross-section shape on the mechanical energy transfer from a uniform flow to a self-excited crossflow oscillating prism. Seventeen cross-section shapes were tested with the same mechanical properties (say, mass ratio and mechanical damping) and aspect ratio. Then, the study was focused on the role of the cross-section shape and reduced velocity on the prism dynamics and energy transfer from the fluid flow. It has been found that the geometric centroid of the semi-cross-section and the afterbody length of the cross-section represent a simple and effective metric to characterise the energy transfer. Two related dimensionless geometric parameters are introduced. The dependence of the energy transfer with these two parameters is studied and used to design successfully ‘in advance’ cross-section shapes with high energy transfer efficiencies. It has been observed from particle image velocimetry measurements that large energy transfer shapes are those in which, under strong oscillations, the flow follows the cross-section contour of one side and is fully detached on the other side.

Experiments have shown that flow in compliant microchannels can become unstable at a much lower Reynolds number than the corresponding flow in a rigid conduit. Therefore, it has been suggested that the wall's elastic compliance can be exploited towards new modalities of microscale mixing. While previous studies mainly focused on the local instability induced by the fluid–structure interactions (FSIs) in the system, we derive a one-dimensional (1-D) model to study the FSI's effect on the global instability. The proposed 1-D FSI model is tailored to long, shallow rectangular microchannels with a deformable top wall, similar to the experiments. Going beyond the usual lubrication flows analysed in these geometries, we include finite fluid inertia and couple the reduced flow equations to a novel reduced 1-D wall deformation equation. Although a quantitative comparison with previous experiments is difficult, the behaviours of the proposed model show, qualitatively, agreement with the experimental observations, and capture several key effects. Specifically, we find the critical conditions under which the inflated base state of the 1-D FSI model is linearly unstable to infinitesimal perturbations. The critical Reynolds numbers predicted are in agreement with experimental observations. The unstable modes are highly oscillatory, with frequencies close to the natural frequency of the wall, suggesting that the observed instabilities are resonance phenomena. Furthermore, during the start-up from an undeformed initial state, self-sustained oscillations can be triggered by FSI. Our modelling framework can be applied to other microfluidic systems with similar geometric scale separation under different operating conditions.

This paper describes, from a theoretical point of view, the appearance and characteristics of granular roll waves in chute flow, and the maximal size these waves can attain for a given influx of material into the system. Granular roll waves are steady travelling wave solutions of the generalized Saint-Venant equations for flowing granular matter, appearing when the Froude number $Fr$ of the incoming flow exceeds a critical value, $Fr>Fr_{cr}$. We focus upon the phase space of the corresponding dynamical system, where the roll waves take the form of a stable limit cycle around an unstable fixed point; this limit cycle gives precise information on the size and periodicity of the roll wave. It is found that, for any given value of $Fr$, the limit cycle cannot become arbitrarily large because it is constrained by a homoclinic loop in phase space. Roll waves of larger amplitude can be generated by increasing the Froude number $Fr$.

We study swirling electrovortex flows in a cylinder filled with GaInSn metal using axisymmetric and large-scale three-dimensional numerical simulations. In our set-up electrical currents enter and exit the cell symmetrically through wires and the result is a von Kármán-like flow. Three inductionless and an inductive flow regimes are identified. Scaling laws for the magnitude of the velocity in each of these regimes are obtained both numerically and explained theoretically. We study how the aspect ratio of the cell affects the flow and how symmetrically wired cells are different from asymmetrical wired cells. We vary the radius of the connecting wires and propose a simple model that captures how the flow's intensity varies with the wire radius.

We study the dynamics of an initially axisymmetric and vertical Lamb–Oseen vortex in a stratified-rotating fluid under the complete Coriolis force on the $f$ plane, i.e. in the presence of a background rotation both along the vertical and horizontal directions. By a combination of direct numerical simulations and asymptotic analyses for small horizontal background rotation, we show that a critical layer appears at the radius where the angular velocity of the vortex is equal to the buoyancy frequency when the Froude number is larger than unity. This critical layer generates a vertical velocity that is invariant along the vertical and which first increases linearly with time and then saturates at an amplitude scaling like $Re^{1/3}/\widetilde{Ro}$, where $Re$ is the Reynolds number and $\widetilde {Ro}$ the non-traditional Rossby number based on the horizontal component of the background rotation. In turn, a quasi-axisymmetric anomaly of vertical vorticity is produced at the critical radius through the non-traditional Coriolis force. Below a critical $\widetilde {Ro}$ depending on $Re$, the Rayleigh's inflectional criterion is satisfied and a shear instability is subsequently triggered rendering the vertical vorticity fully non-axisymmetric. The decay of the angular velocity is then enhanced until it is everywhere lower than the buoyancy frequency. A theoretical criterion derived from the Rayleigh condition predicts well the instability. It shows that this phenomenon can occur even for a large non-traditional Rossby number $\widetilde {Ro}$ for large $Re$. Hence, the non-traditional Coriolis force might have much more impact on geophysical vortices than anticipated by considering the order of magnitude of $\widetilde {Ro}$.

We investigate the incompressible flow past a square cylinder immersed in the wake of an upstream nearby splitter plate separating two streams of different velocity. The bottom stream Reynolds number, based on the square side, $Re_B=56$ is kept constant while the top-to-bottom Reynolds numbers ratio $R\equiv Re_T/Re_B$ is increased in the range $R\in [1,6.5]$, corresponding to a coupled variation of the bulk Reynolds number $Re\equiv (Re_T+Re_B)/2\in [56,210]$ and an equivalent non-dimensional shear parameter $K\equiv 2(R-1)/(R+1)\in [0,1.4667]$. The onset of vortex shedding is pushed to higher $Re$ as compared with the square cylinder in the classic configuration. The advent of three dimensionality is triggered by a mode-C-type instability, much as reported for open circular rings and square cylinders placed at an incidence. The domain of minimal spanwise-periodic extension that is capable of sustaining spatiotemporally chaotic dynamics, namely one accommodating about twice the wavelength of the dominant eigenmode, has been chosen for the analysis of the wake transition regime. The path towards spatiotemporal chaos is, in this minimal domain, initiated with a modulational period-doubling tertiary bifurcation that also doubles the spanwise periodicity. At slightly higher values of $R$, the flow has become spatiotemporally chaotic, but the main features of mode C are still clearly distinguishable. Although some of the nonlinear solutions found along the wake transition regime employing the minimal domain might indeed be unstable to long wavelength disturbances, they still are solutions of the infinite cylinder problem and are apt to play a relevant role in the inception of spatiotemporally chaotic dynamics.

The wake of a wall-mounted finite prism is numerically studied and characterized with an aspect ratio (height-to-width) of $1$ and varying depth ratios (length to width) of between $0.016$ and $4$ at Reynolds numbers of 50–500. The prism is immersed in a laminar boundary layer. The minimum depth ratio considered here accounts for the special case of a wall-mounted very thin prism (similar to a flat plate), which is used to establish the mechanism and evolution of the wake associated with free-end effects and the shear-layer dynamics in small aspect-ratio prisms. The onset of an unsteady wake behind a very thin prism at a Reynolds number of $200$ is characterized by symmetric shedding of hairpin-like vortices. A unique asymmetric wake pattern appears at lower depth ratios starting at a Reynolds number of 250, which transitions to an symmetric wake with increasing depth ratio. The threshold depth ratio for this symmetric transition increases with Reynolds number. The asymmetric wake results from alternate shear-layer peel-off from either side of the prism, which itself is attributed to the out-of-phase shedding of tip vortices at a lower Strouhal number ($St_{sh}/2$) that interact with the detaching side shear layers. Alternate shedding of tip vortices form secondary vortex structures that are fed by the excess vorticity resulting from shear-layer detachment from either side of the prism. Increasing the depth ratio leads to simultaneous shedding of the tip vortices, which restores the commonly observed wake symmetric patterns. Thus, we identify and characterize the formation and interaction mechanisms of symmetric and asymmetric wakes during the transition process with increasing Reynolds number for different depth-ratio prisms.

The effect of a zero-mean-flow turbulent ambient on the geometry of the turbulent/turbulent interface (TTI) of an axisymmetric jet is investigated and compared with the traditional turbulent/non-turbulent interface (TNTI). The ambient turbulence is generated using a random jet array. Orthogonal cross-sections of the jet subjected to different levels of ambient turbulence intensities and length scales are captured using planar laser-induced fluorescence. The enhanced radial transport of concentration from the jet core towards the interface and the increased scalar fluctuations within the jet caused by the ambient turbulence result in steeper mean and root-mean-square scalar conditional jumps across the TTI layer compared with those of the TNTI, respectively. When compared with the quiescent ambient, the mean effect of background turbulence is to stretch and corrugate the interface surface area as evident from the wider probability density of the interface radial position, the lowered occurrence of zero-curvature surface elements, the greater misalignment between the radial and normal unit vectors of the interface, the increased tortuosity and the increased magnitude of the fractal exponent.

An explicit expression for the Dirichlet–Neumann operator for surface water waves is presented. For non-overturning waves, but without assuming small amplitudes, the formula is first derived in two dimensions, and subsequently extrapolated to higher dimensions and with a moving bottom. Although described here for water waves, this elementary approach could be adapted to many other problems having similar mathematical formulations.

This paper revisits the melting process of phase change materials (PCMs) enclosed in a horizontally placed rectangular cavity, with isothermal and adiabatic conditions subjected to the vertical and horizontal walls, respectively. First, numerical simulations based on an improved lattice Boltzmann method are conducted to illustrate and to inform the theoretical modelling. It is shown that, compared with the traditional two-stage conduction–convection melting description, it is more reasonable to include a third stage in terms of the heat transfer behaviour. During the third stage, the remnant solid PCM is located in the corner formed by the cold and bottom walls of the cavity, and an increasing part of the input energy will be transferred directly out of the cavity without compensating for the melting latent heat, thus inducing a continuously decreasing melting rate until the end of the melting process. Then theoretical predictions are derived piecewise for the melted liquid fraction during the entire melting process, and the corresponding transitions between two successive stages are also discussed. The results are validated successfully via the available experimental and numerical data in the literature, and could guide the design and operation of latent heat storage systems.