The shape of a free-surface slump of viscoplastic material supported by an oblique barrier on an inclined plane is investigated theoretically and experimentally. The barrier is sufficiently tall that it is not surmounted by the viscoplastic fluid, and a focus of this study is the largest volume of rigid viscoplastic fluid that can be supported upstream of it. A lubrication model is integrated numerically to determine the transient flow as the maximal rigid shape is approached. Away from the region supported by the barrier, the viscoplastic layer attains a uniform thickness in which the gravitational stresses are in balance with the yield stress of the material. However, closer to the barrier, the layer thickens and the barrier bears the additional gravitational loading. An exact solution for the rigid shape of the viscoplastic material is constructed from the steady force balance and computed by integrating Charpit’s equations along characteristics that emanate from the barrier wall. The characteristics represent the late-time streamlines of the flow as it approaches the rigid shape. The exact solution depends on a single dimensionless group, which incorporates the slope inclination, the barrier width and the fluid’s yield stress. It is shown that the shape is insensitive to the transient flow from which it originates. The force exerted by the slump is calculated for different barrier shapes. The results of new laboratory experiments are reported; these show that although convergence to the final rigid state is slow, there is good agreement with the experimental measurements at long times.

The pressure effects on the mixing fields of non-reacting and reacting jets in cross-flow are studied using large eddy simulation (LES). A hydrogen jet diluted with 30 % helium is injected perpendicularly into a cross-stream of air at four different pressures: 1, 4, 7 and 15 bar. The resulting interaction and the mixing fields under non-reacting and reacting conditions are simulated using LES. The subgrid scale combustion is modelled using a revised flamelet model for the partially premixed combustion. Good agreement of computed and measured velocity fields for reacting and non-reacting conditions is observed. Under non-reacting conditions, the mixing field shows no sensitivity to the pressure, whereas notable changes are observed for reacting conditions. The lifted flame at 1 bar moves upstream and attaches to the nozzle as the pressure is increased to 4 bar and remains so for the other elevated pressures because of the increasing burning mass flux with pressure. This attached flame suppresses the fuel–air mixing in the near-nozzle region. The premixed and non-premixed contributions to the overall heat release in the partially premixed combustion are analysed. The non-premixed contribution is generally low and occurs in the near-field region of the fuel jet through fuel-rich mixtures in the shear layer regions, and decreases substantially further with the increase in pressure. Hence, the predominant contributions are observed to come from premixed modes and these contributions increase with pressure.

Recent studies reveal the central role of chaotic advection in controlling pore-scale processes including solute mixing and dispersion, chemical reactions, and biological activity. These dynamics have been observed in porous media (PM) with a continuous solid phase (such as porous networks) and PM comprising discrete elements (such as granular matter). However, a unified theory of chaotic advection across these continuous and discrete classes of PM is lacking. Key outstanding questions include: (i) topological unification of discrete and continuous PM; (ii) the impact of the non-smooth geometry of discrete PM; (iii) how exponential stretching arises at contact points in discrete PM; (iv) how fluid folding arises in continuous PM; (v) the impact of discontinuous mixing in continuous PM; and (vi) generalised models for the Lyapunov exponent in both PM classes. We address these questions via a unified theory of pore-scale chaotic advection. We show that fluid stretching and folding (SF) in discrete and continuous PM arise via the topological complexity of the medium. Mixing in continuous PM manifests as discontinuous mixing through a combination of SF and cutting and shuffling (CS) actions, but the rate of mixing is governed by SF only. Conversely, discrete PM involves SF motions only. These mechanisms are unified by showing that continuous PM is analogous to discrete PM with smooth, finite contacts. This unified theory provides insights into the pore-scale chaotic advection across a broad class of porous materials and points to design of novel porous architectures with tuneable mixing and transport properties.

A block of ice in a box heated from below and cooled from above can (partially) melt. Vice versa, a box of water with less heating from below or more cooling from above can (partially) re-solidify. This study investigates the asymmetric behaviours between such melting and freezing processes in this Rayleigh–Bénard geometry, focusing on differences in equilibrium flow structures, solid–liquid interface morphology, and equilibrium mean interface height. Our findings reveal a robust asymmetry across a range of Rayleigh numbers and top cooling temperature (i.e. hysteretic behaviour), where the evolution of freezing shows a unique ‘splitting event’ of convection cells that leads to a non-monotonic height evolution trend. To characterise the differences between melting and freezing, we introduce an effective Rayleigh number and the aspect ratio for the cellular structures, and apply the heat flux balance and the Grossmann–Lohse theory. Based on this, we develop a unifying model for the melting and freezing behaviour across various conditions, accurately predicting equilibrium states for both phase-change processes. This work provides insights into the role of convective dynamics in phase-change symmetry-breaking, offering a framework applicable to diverse systems involving melting and freezing.

Buoyancy-driven exchange flows in geophysical contexts often exhibit significant interfacial turbulence leading to a partially mixed intermediate layer between two counterflowing layers. In this paper we perform a three-layer hydraulic analysis of such flows, highlighting the dynamical importance of the middle mixed layer. Our analysis is based on the viscous, shallow water, Boussinesq equations and includes the effects of mixing as a non-hydrostatic pressure forcing. We demonstrate the superior predictive accuracy of three-layer hydraulics over the more classical two-layer approach by applying it to direct numerical simulation data in stratified inclined duct exchange flows where turbulence is controlled by a modest slope of the duct. The three-layer model predicts a region bounded by two control points in the middle of the duct, linked to the onset of instability and turbulence, whereas a two-layer model only predicts one control point. We show that the nonlinear characteristics of the three-layer model correspond to linear long waves perturbing a three-layer mean flow. We also provide the first evidence of long-wave resonance, as well as resonance between long and short waves, and their connection to turbulence. These results challenge current parameterisations for turbulent transport, which typically overlook long waves and internal hydraulics induced by streamwise variations of the flow.

We focus on the wake of a cylinder placed in uniform flow and forced to rotate periodically at subcritical Reynolds numbers, i.e. for Reynolds numbers smaller than 47 calculated based on the incoming flow velocity and the cylinder diameter, where vortices are not shed in the wake of a fixed cylinder. We show that in the near wake, the imposed periodic rotation causes the Föppl vortices (the symmetric steady vortices that are formed right behind a fixed cylinder within the Reynolds number range of $5\lt {Re}\lt 47$) to appear only momentarily during each rotation cycle until they disappear at higher rotation rates. In the far wake, vortices can be induced for certain values of rotation rate, $\alpha$, and rotation frequency, $f$. The shedding of these vortices in the wake results in a periodic lift force that acts on the cylinder. We have defined a new parameter $\omega /(f\alpha )\equiv 1/F$, where $\omega$ is the angular velocity of the cylinder, which is significant in describing the system. For any values of angular velocity and the frequency of change in the rotation direction, the wake pattern remains the same if the value of $1/F$ stays constant. Subsequently, the fluctuating lift coefficient and the average drag coefficient peak at the same value of $1/F$ for any value of $\omega /f\equiv \alpha /F$. The Reynolds number for the onset of shedding decreases with increasing rotation rate at a constant $\alpha /F$. We have observed shedding at Reynolds numbers as low as ${Re}=1$ for higher rotation rates.

The reactivity of transverse waves in detonations of methane, oxygen and nitrogen are experimentally assessed using MHz rate schlieren and chemiluminescence imaging. In these highly unstable mixtures, the mode of wave propagation is more complex than what is described by the cellular instability model that is conventionally used for weakly unstable mixtures. Behind the low-speed leading shock in unstable waves, the processed gas remains essentially unreacted until transverse waves reach this region. In highly unstable waves, the transverse waves have a range of reactivity, that is rates of reaction in the flow immediately behind the wave. In this study, we present examples of transverse waves for near-limit detonations and analyse four cases in detail. In some cases, these waves appear to be essentially non-reactive or cause very slow reaction. In other cases, the transverse waves can be highly reactive. In the most extreme example, the transverse wave is propagating at the Chapman–Jouguet speed with a small reaction zone, i.e. a transverse detonation. A reactive oblique shock model is used to approximate the triple-point configuration of this case as a double-Mach reflection, which shows good agreement with the images. The reaction evolution along path lines is analysed using detailed reaction mechanisms and considerations about flow-field unsteadiness. Length scales of the energy release and expansion processes within the reaction zone region are used to explain the observed modes of wave propagation and interaction.

We address the Reynolds number dependence of the turbulent skin-friction drag reduction induced by streamwise-travelling waves of spanwise wall oscillations. The study relies on direct numerical simulations of drag-reduced flows in a plane open channel at friction Reynolds numbers in the range $1000 \leqslant Re_\tau \leqslant 6000$, which is the widest range considered so far in simulations with spanwise forcing. Our results corroborate the validity of the predictive model proposed by Gatti & Quadrio (J. Fluid Mech. vol. 802, 2016, pp. 553–558): regardless of the control parameters, the drag reduction decreases monotonically with $Re$ at a rate that depends on the drag reduction itself and on the skin-friction of the uncontrolled flow. We do not find evidence in support of the results of Marusic et al. (Nat. Commun. vol. 12, no. 1, 2021, pp. 5805), which instead report by experiments an increase of the drag reduction with $Re$ in turbulent boundary layers, for control parameters that target low-frequency, outer-scaled motions. Possible explanations for this discrepancy are provided, including obvious differences between open channel flows and boundary layers, and possible limitations of laboratory experiments.

Following Scott & Cambon (2024 J. Fluid Mech. vol. 979, A17), henceforth referred to as [I], a spectral approach is used and the flow is expressed as a sum of normal modes, which are of two types: inertial/gravity waves and non-propagating (NP) modes. It was shown in [I] that, for weak (small Rossby or Froude number) turbulence, the NP component of the flow decouples from the waves at leading order and here we focus on the NP part alone. It is demonstrated that the evolution equations of the NP component are equivalent to the three-dimensional, quasi-geostrophic (QG) approximation of geophysical fluid dynamics. For QG turbulence, the seminal paper of Charney (1971 J. Atmos. Sci. vol. 28, pp. 1087–1095), referred to as [II], concluded that, as for two-dimensional turbulence, the energy cascade for QG turbulence should go from smaller to larger scales and that the inertial-range spectrum at wavenumber $k$ should behave as $k^{-3}$. He also proposed that the energy distribution in spectral space is isotropic if the vertical wavenumber is appropriately scaled and deduced a principle of equipartition in which the average kinetic energy is twice the potential one. We use Charney’s transformation of spectral coordinates to effectively eliminate the parameter $\beta =2{\varOmega} /N$, where ${\varOmega}$ is the rotation rate and $N$ the Brunt–Vaisala frequency, and give results of numerical calculations concerning the energy distribution. The results mostly agree with [II] at large enough times, although they do not support Charney isotropy. They further suggest self-similarity of the time evolution of the three-dimensional energy distribution in spectral space away from the vertical axis.

This paper presents an experimental application of reactive control to jet installation noise based on destructive interference. The work is motivated by the success of previous studies in applying this control approach to mixing layers (Sasaki et al. Theor. 2018b Comput. Fluid Dyn. 32, 765–788), boundary layers (Brito et al. 2021 Exp. Fluids 62, 1–13; Audiffred et al. 2023 Phys. Rev. Fluids 8, 073902), flow over a backward-facing step (Martini et al. 2022 J. Fluid Mech. 937, A19) and, more recently, to turbulent jets (Maia et al. 2021 Phys. Rev. Fluids 6, 123901; Maia et al. 2022 Phys. Rev. Fluids 7, 033903; Audiffred et al. 2024b J. Fluid Mech. 994, A15). We exploit the fact that jet–surface interaction noise is underpinned by wavepackets that can be modelled in a linear framework and develop a linear control strategy where piezoelectric actuators situated at the edge of a scattering surface are driven in real time by sensor measurements in the near field of the jet, the objective being to reduce noise radiated in the acoustic field. The control mechanism involves imposition of an anti-dipole at the trailing edge to cancel the scattering dipole that arises due to an incident wavepacket perturbation. We explore two different control strategies: (i) the inverse feed-forward approach, where causality is imposed by truncating the control kernel, and (ii) the Wiener–Hopf approach, where causality is optimally enforced in building the control kernel. We show that the Wiener–Hopf approach has better performance than that obtained using the truncated inverse feed-forward kernel. We also explore different positions of the near-field sensors and show that control performance is better for sensors installed for streamwise positions downstream in the jet plume, where the signature of hydrodynamic wavepacket is better captured by the sensors. Broadband noise reductions of up to 50 % are achieved.

We study the mixing of passive scalars in a velocity field generated by selected-eddy simulations (SES), an approach where only a randomly selected subset of spectrally distributed modes obey Navier–Stokes dynamics. The Taylor Reynolds number varies from 140 to 400 and the Schmidt number ($Sc$) varies from 0.25 to 1. By comparing the results with direct numerical simulations (DNS), we show that most statistics are captured with as low as $0.5\,\%$ of Navier–Stokes modes in the velocity field. This includes scalar gradients, spectra, structure functions and their departures from classical scaling due to intermittency. The results suggest that all modes need not be resolved to accurately capture turbulent mixing for $Sc\leqslant 1$ scalars.

The evaporation of liquid from within a porous medium is a complicated process involving coupled capillary flow, vapour diffusion and phase change. Different drying behaviour is observed at different stages during the process. Initially, liquid is drawn to the surface by capillary forces, where it evaporates at a near constant rate; thereafter, a drying front recedes into the material, with a slower net evaporation rate. Modelling drying porous media accurately is challenging due to the multitude of relevant spatial and temporal scales, and the large number of constitutive laws required for model closure. Key aspects of the drying process, including the net evaporation rate and the time of the sudden transition between stages, are not well understood or reliably predicted. We derive simplified mathematical models for both stages of this drying process by systematically reducing an averaged continuum multi-phase flow model, using the method of matched asymptotic expansions, in the physically relevant limit of slow vapour diffusion relative to the local evaporation rate (the large-Péclet-number limit). By solving our reduced models, we compute the evolving net evaporation rate, fluid fluxes and saturation profiles, and estimate the transition time to be when the initial constant-rate-period model ceases to be valid. We additionally characterise properties of the constitutive laws that affect the qualitative drying behaviour: the model is shown to exhibit a receding-front period only if the relative permeability for the liquid phase decays sufficiently quickly relative to the blow up in the capillary pressure as the liquid saturation decreases.

Understanding how bubbles on a substrate respond to ultrasound is crucial for applications from industrial cleaning to biomedical treatments. Under ultrasonic excitation, bubbles can undergo shape deformations due to Faraday instability, periodically producing high-speed jets that may cause damage. While recent studies have begun to elucidate this behaviour for free bubbles, the dynamics of wall-attached bubbles is still largely unexplored. In particular, the selection and evolution of non-spherical modes in these bounded systems have not previously been resolved in three dimensions, and the resulting jetting dynamics has yet to be compared with that observed in free bubbles. In this study, we investigate individual micrometric air bubbles in contact with a rigid substrate and subjected to ultrasound. We introduce a novel dual-view imaging technique that combines top-view bright-field microscopy with side-view phase-contrast X-ray imaging, enabling visualisation of bubble shape evolution from two orthogonal perspectives. This set-up reveals the progression of bubble shape through four distinct dynamic regimes: purely spherical oscillations, onset of harmonic axisymmetric meniscus waves, emergence of half-harmonic axisymmetric Faraday waves and the superposition of half-harmonic sectoral Faraday waves. This stepwise evolution contrasts with the behaviour of free bubbles, which exhibit their ultimate Faraday wave pattern immediately upon instability onset. For the substrate chosen, the resulting shape-mode spectrum appears to be degenerate and exhibits a continuous range of shape mode degrees, in line with our theoretical predictions derived from kinematic arguments. While free bubbles also display a degenerate spectrum, their shape mode degrees remain discrete, constrained by the bubble spherical periodicity. Experimentally measured ultrasound pressure thresholds for the onset of Faraday instability agree well with classical interface stability theory, modified to incorporate the effects of a rigid boundary. Complementary three-dimensional boundary element simulations of bubble shape evolution align closely with experimental observations, validating this method’s predictive capability. Finally, we determine the acceleration threshold at which shape mode lobes initiate cyclic jetting. Unlike free bubbles, jetting in wall-attached bubbles consistently emerges from the side not restricted by the substrate.

Suspensions of microswimmers exhibit distinct characteristics as compared with those of passive particles because the internal particles are in a state of spontaneous motion. Although there have been many studies of microswimmer suspensions, not many have carefully considered the hydrodynamics. Hydrodynamics becomes particularly important when discussing non-dilute suspensions, because the lubrication flow generates a large force when the swimmers are in close proximity. This paper focuses on hydrodynamics and describes the transport phenomena of microswimmer suspensions, such as migration, collective motion, diffusion and rheology. The paper is structured to progressively scale up from a single microswimmer to collective motion to a macroscale continuum. At each scale, the discussion also evolves from dilute to concentrated suspensions. We first introduce natural swimming microorganisms, artificial microswimmers and mathematical models, as well as the fundamentals of fluid mechanics relevant to microswimmers. We then describe the migration of microswimmers by taxis, where microswimmers respond passively or actively to their hydrodynamic environment. Microswimmers exhibit collective motions, the mechanism of which is discussed in terms of hydrodynamics. The spreading of microswimmers is often diffusive, and the diffusion coefficient is much larger than for passive particles. Similarly, the mass diffusivity in microswimmer suspensions is higher due to their swimming activity. We explain these macroscopic diffusion properties. The viscosity of microswimmer suspensions can be higher or lower depending on the characteristics and orientation of the microswimmers. We describe the rheological properties of microswimmer suspensions in shear flow and Poiseuille flow. Finally, current issues and future research perspectives are discussed.

Direct numerical simulation (DNS) of a Mach 4.9 zero-pressure-gradient turbulent boundary layer spatially developing over a cooled flat plate at wall-to-recovery temperature $T_w/T_r = 0.60$ is performed. Very long, streamwise contiguous domains are used in the DNS to achieve a wide continuous range of ‘useful’ friction Reynolds numbers of $1000 \lesssim {Re}_\tau \lesssim 2500$. The DNS datasets have been analysed to assess state-of-the-art compressibility scaling relations and turbulence modelling assumptions. The DNS data show a notable distinction in Reynolds number dependence between thermal and velocity fields. Although Reynolds stress and the budgets of turbulent kinetic energy have reached Reynolds number independence in the inner layer under semi-local scaling by ${Re}_\tau \simeq 1000$, the budget terms for temperature variance and turbulent heat flux retain a clear Reynolds number dependence near the wall over a broader range up to ${Re}_\tau \simeq 1900$. Such a stronger dependence of the thermal field on the Reynolds number may lead to inaccuracy in turbulence models that are calibrated on the basis of low-Reynolds-number data. Spectral and structural analysis suggests a more significant reduction in the prevalence of alternating positive and negative structures and an increase in the streamwise uniformity of streaks in the wall heat flux $q_w$ than in the wall shear stress $\tau _w$ when the Reynolds number increases.

The impact of two-dimensional (2-D) periodic forcing on transition dynamics in laminar separation bubbles (LSBs) generated on a flat plate is investigated experimentally. Laminar separation is caused by the favourable-to-adverse pressure gradient under an inverted modified NACA $64_3\text{-}618$ and periodic disturbances are generated by an alternating current dielectric barrier discharge plasma actuator located near the onset of the adverse pressure gradient. Surface pressure and time-resolved particle image velocimetry measurements along the centreline and several wall-parallel planes show significant reductions in bubble size with active flow control. Periodic excitation leads to amplification of the Kelvin–Helmholtz (K–H) instability resulting in strong 2-D coherent roller structures. Spanwise modulation of these structures is observed and varies with the forcing amplitude. Intermediate forcing amplitudes result in periodic spanwise deformation of the mean flow at large wavelength ($\lambda _z/L_{b,5kVpp} \approx 0.76$). For high-amplitude forcing, the spanwise modulation of the mean flow agrees with the much smaller wavelength of the difference interaction of two oblique subharmonic modes ($\lambda _z/L_{b,5kVpp} \approx 0.24$). Modal decomposition shows nonlinear interaction of the forced 2-D mode leading to growth of subharmonic and harmonic content, and the observation of several half-harmonics ($[n+1/2]f_{\textit{AFC}}$) at intermediate forcing amplitudes. Strongest amplitudes of the 2-D mode and delay of transition downstream of the time-averaged reattachment are observed for the intermediate forcing amplitudes, previously only observed in numerical simulations. Consistent with numerical results, further increase of the forcing amplitude leads to rapid breakdown to turbulence in the LSB. This suggests that the most effective exploitation of the K–H instability for transition delay is connected to an optimal (moderate) forcing amplitude.

In gas evolving electrolysis, bubbles grow at electrodes due to a diffusive influx from oversaturation generated locally in the electrolyte by the electrode reaction. When considering electrodes of micrometre size resembling catalytic islands, direct numerical simulations show that bubbles may approach dynamic equilibrium states at which they neither grow nor shrink. These are found in undersaturated and saturated bulk electrolytes during both pinning and expanding wetting regimes of the bubbles. The equilibrium is based on the balance of local influx near the bubble foot and global outflux. To identify the parameter regions of bubble growth, dissolution and dynamic equilibrium by analytical means, we extend the solution of Zhang & Lohse (2023 J. Fluid Mech. vol. 975, R3) by taking into account modified gas fluxes across the bubble interface, which result from a non-uniform distribution of dissolved gas. The Damköhler numbers at equilibrium are found to range from small to intermediate values. Unlike pinned nanobubbles studied earlier, for micrometre-sized bubbles the Laplace pressure plays only a minor role. With respect to the stability of the dynamic equilibrium states, we extend the methodology of Lohse & Zhang (2015a Phys. Rev. E vol. 91, 031003(R)) by additionally taking into account the electrode reaction. Under contact line pinning, the equilibrium states are found to be stable for flat nanobubbles and for microbubbles in general. For unpinned bubbles, the equilibrium states are always stable. Finally, we draw conclusions on how to possibly enhance the efficiency of electrolysis.

We investigate flow-induced choking in soft Hele-Shaw cells comprising a fluid-filled gap in between a rigid plate and a confined block of elastomer. Fluid injected from the centre of the circular rigid plate flows radially outwards, causing the elastomeric block to deform, before exiting through the cell rim. The pressure in the fluid deforms the elastomer, increasing the size of the gap near the inlet, and decreasing the gap near the cell rim, because of volume conservation of the solid. At a critical injection flow rate, the magnitude of the deformation becomes large enough that the flow is occluded entirely at the rim. Here, we explore the influence of elastomer geometry on flow-induced choking and, in particular, the case of a thick block with radius smaller than its depth. We show that choking can still occur with small-aspect-ratio elastomers, even though the confining influence of the back wall that bounds the elastomer becomes negligible; in this case, the deformation length scale is set by the radial size of the cell rather than the depth of the block. Additionally, we reveal a distinction between flow-induced choking in flow-rate-controlled flows and flow-rate-limiting behaviour in pressure-controlled flows.