Melting and solidification in periodically time-modulated thermal convection are relevant for numerous natural and engineering systems, for example, glacial melting under periodic sun radiation and latent thermal energy storage under periodically pulsating heating. It is highly relevant for the estimation of melt rate and melt efficiency management. However, even the dynamics of a solid–liquid interface shape subjected to a simple sinusoidal heating has not yet been investigated in detail. In this paper, we offer a better understanding of the modulation frequency dependence of the melting and solidification front. We numerically investigate periodic melting and solidification in turbulent convective flow with the solid above and the melted liquid below, and sinusoidal heating at the bottom plate with the mean temperature equal to the melting temperature. We investigate how the periodic heating can prevent the full solidification, and the resulting flow structures and the quasi-equilibrium interface height. We further study the dependence on the heating modulation frequency. As the frequency decreases, we found two distinct regimes, which are ‘partially solid’ and ‘fully solid’. In the fully solid regime, the liquid freezes completely, and the effect of the modulation is limited. In the partially solid regime, the solid partially melts, and a steady or unsteady solid–liquid interface forms depending on the frequency. The interface height can be derived based on the energy balance through the interface. In the partially solid regime, the interface height oscillates periodically, following the frequency of modulation. Here, we propose a perturbation approach that can predict the dependency of the oscillation amplitude on the modulation frequency.

Microvortex generators (MVGs) are a promising solution to control shock wave/turbulent boundary layer interactions (SBLIs), especially in supersonic inlets. In this study, we examine the effects of a microramp vortex generator on an SBLI generated by an oblique shock wave and a turbulent boundary layer using direct numerical simulations (DNSs). Two cases, with and without the presence of a microramp, are compared in terms of their mean and unsteady flow features at free-stream Mach number equal to 2 and friction Reynolds number at the inviscid shock impingement equal to 600. The long integration period allows us to assess how microramps affect the typical low-frequency unsteadiness observed in SBLIs, and the data generated may serve as a reference for simulations of lower fidelity or reduced order models. The analysis shows that the three-dimensional microramp wake alters the interaction region dramatically, inducing a significant spanwise modulation and topology change of the separation. For example, tornado-like structures redistribute the flow in both the spanwise and wall-normal directions inside the recirculation region. The increase in momentum close to the wall by the ramp vortices effectively delays the onset of the separation and, thus, the separation length, but at the same time leads to a significant increase in the intensity of the wall-pressure fluctuations. We then characterise the mutual interaction between the arch-like vortices around the ramp wake and the SBLI. The specific spanwise vorticity shows that these vortices follow the edge of the separation and their intensity, apart from mean compressibility effects, is not affected by the shocks. The shocks, instead, are deformed in shape by the periodic impingement of the vortices, although the spectral analysis did not reveal any significant trace of their shedding frequency in the separation region. These Kelvin–Helmholtz vortices, however, may be relevant in the closure of the separation bubble. Fourier analysis also shows a constant increase, in both value and magnitude, in the low-frequency peak all along the span, suggesting that the motion of the separation shock remains coherent while being disturbed by the arch-like vortices and oscillating at a higher frequency in absolute terms.

We investigate a turbulent boundary layer (TBL) under uniform pressure force variations, focusing on understanding its response to local pressure force, local pressure force variation (local disequilibrating effect) and upstream history. The flow starts as a zero-pressure-gradient (ZPG) TBL, followed by a uniform increase in the ratio of pressure force to turbulent force in the outer region and concludes with a uniform decrease of the same magnitude. This last zone includes a subzone with a diminishing adverse pressure gradient (APG), followed by an increasing favourable pressure gradient (FPG). In both subzones, the impact remains the same: mean momentum gain and turbulence reduction. In the outer region, the mean flow responds to force balance changes with a considerable delay. The accumulated flow history leads to a FPG TBL at the domain's end with a momentum defect comparable to APG TBLs. Below $y^+=10$, the mean flow responds almost instantaneously to pressure force changes. In the overlap layer, velocity profiles deviate from the conventional logarithmic law of ZPG TBL. Outer-layer turbulence decays more slowly than it increases initially, the latter turbulence increase persisting even after the pressure force begins to decrease. As a result of the slow turbulence decay, the FPG TBL at the domain's end exhibits unusually high outer turbulence levels. Near the wall, turbulence responds with a delay to pressure force changes, partly due to the influence of large-scale turbulence. All these significant cumulative effects of continuous pressure force variation indicate that parameters based solely on local variables cannot fully describe the physics of non-equilibrium TBLs.

We perform simulations of an impulsively started, axisymmetric viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively refined volume-of-fluid technique that can accurately capture the deformation of the liquid–gas interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the liquid, and employ the log-conformation transformation, which provides stable solutions for the evolution of the conformation tensor as the jet thins down under the action of interfacial tension. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the pre-shearing flow through the nozzle, which results in an inhomogeneous initial radial stress distribution in the fluid thread that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions where possible, and the early-stage dynamics of the jet evolution are compared favourably to the predictions of linear stability theory. We study the effect of the flow inside the nozzle on the thinning dynamics of the viscoelastic jet (which develops distinctive ‘beads-on-a-string’ structures) and on the spatio-temporal evolution of the polymeric stresses in order to systematically explore the dependence of the filament thinning and breakup characteristics on the initial axial momentum of the jet and the extensibility of the dissolved polymer chains.

Arrays of heaving buoy type wave energy converters (WECs) are a promising contender to harness the renewable power of ocean waves on a commercial scale but require strategies to achieve efficient capture of wave energy over broad frequency bands for economic viability. A WEC-array design is proposed for absorption over a target frequency range in the two-dimensional water wave context by spatially grading the resonant properties of WECs via linear spring–damper power take-off mechanisms. The design is based on theories for rainbow reflection and rainbow absorption, which incorporate analyses based on Bloch wave modes and pole–zero pairs in complex frequency space. In contrast to previous applications of these theories, the influence of a higher-order passband and associated pole–zero pairs are shown to influence absorption at the high-frequency end of the target interval. The theories are used to inform initialisations for optimisation algorithms, and an optimised array of only five WECs is shown to give near-perfect absorption ($\geq$99 %) over the target interval. Broadband absorption is demonstrated when surge and pitch motions are released, for irregular sea states, and for incident wave packets in the time domain, where the time-domain responses are decomposed into Bloch modes to connect with the underlying theory.

In the present work, the tangential (swirl) velocity component is superimposed at the intake of a narrow fluidic cylindrical pipe to achieve the desired mixing of inelastic non-Newtonian fluids/solutes at the outlet. We discuss an analytical method for obtaining the swirl velocity profile, considering the nonlinear viscous effects for both shear-thinning and shear-thickening fluids, represented by the power-law model. We numerically solve the species transport equation, coupled with the analytically derived swirl velocity, using our in-house developed code for the concentration distribution in the flow field. The results show that the inlet swirl and an increase in the shear-thinning fluid property improve advection-dominated mixing. Additionally, higher Reynolds numbers significantly enhance advection's dominance, as more rotation leads to the generation of vortices, resulting in an engulfment flow (chaotic convection) based mixing. We demonstrate that considering the increase in the shear-thinning fluid property with swirl intake reduces the amount of mixing time required in the convective regime.

The buoyancy-driven motion of a deformable bubble rising near a vertical hydrophilic wall is studied numerically. We focus on moderately inertial regimes in which the bubble undergoes low-to-moderate deformations and would rise in a straight line in the absence of the wall. Three different types of near-wall motion are observed, depending on the buoyancy-to-viscous and buoyancy-to-capillary force ratios defining the Galilei ($Ga$) and Bond ($Bo$) numbers of the system, respectively. For low enough $Ga$ or large enough $Bo$, bubbles consistently migrate away from the wall. Conversely, for large enough $Ga$ and low enough $Bo$, they perform periodic near-wall bounces. At intermediate $Ga$ and $Bo$, they are first attracted to the wall down to a certain critical distance, and then perform bounces with a decreasing amplitude before stabilising at this critical separation. Periodic bounces are accompanied by the shedding of a pair of streamwise vortices in the wake, the formation of which is governed by the near-wall shear resulting from the no-slip condition. These vortices provide a repulsive force that overcomes the viscous resistance of the fluid to the departing motion, making the bubble capable of returning to the region where it is attracted again to the wall. Although periodic, the shedding/regeneration cycle of these vortices is highly asymmetric with respect to the lateral bubble displacements, vortices being shed when the gap left between the bubble and the wall reaches its maximum, and reborn only when this gap comes back to its minimum.

In an open channel flow, deviations to the lower topography can induce abrupt changes in the wave height, known as hydraulic jumps. This phenomenon occurs when the flow switches from subcritical to supercritical (or vice versa), and is commonly observed in rivers, flumes and weirs. Theoretical insight is typically sought through the study of reduced models such as the forced Korteweg–de Vries equation, in which previous work has predominantly focused on either stationary formulations or the initial transient behaviour caused by perturbations. In a joint theoretical and numerical study of the free-surface Euler equations, Keeler & Blyth (J. Fluid Mech., vol. 993, 2024, A9) have detected a new class of unsteady solutions to this problem. These emerge from an unstable steady solution, and feature large-amplitude time-periodic ripples emitted from a sudden decrease in the water depth forced by topography, known as a hydraulic fall.

Free surface flows driven by boundary undulations are observed in many biological phenomena, including the feeding and locomotion of water snails. To simulate the feeding strategy of apple snails, we develop a centimetric robotic undulator that drives a thin viscous film of liquid with the wave speed $V_w$. Our experimental results demonstrate that the behaviour of the net fluid flux $Q$ strongly depends on the Reynolds number $Re$. Specifically, in the limit of vanishing $Re$, we observe that $Q$ varies non-monotonically with $V_w$, which has been successfully rationalised by Pandey et al. (Nat. Commun., vol. 14, no. 1, 2023, p. 7735) with the lubrication model. By contrast, in the regime of finite inertia (${Re} \sim O(1)$), the fluid flux continues to increase with $V_w$ and completely deviates from the prediction of lubrication theory. To explain the inertia-enhanced pumping rate, we build a thin-film, two-dimensional model via the asymptotic expansion in which we linearise the effects of inertia. Our model results match the experimental data with no fitting parameters and also show the connection to the corresponding free surface shapes $h_2$. Going beyond the experimental data, we derive analytical expressions of $Q$ and $h_2$, which allow us to decouple the effects of inertia, gravity, viscosity and surface tension on free surface pumping over a wide range of parameter space.

In this study, the resolvent-based estimation (RBE) is further generalised to cases with arbitrarily sampled measurements in time, where the generalised RBE is denoted as GRBE in this study. Different from the RBE that constructs the transfer function at each frequency, the GRBE minimises the estimation error energy in the physical temporal domain by considering the forcing and noise statistics. The GRBE is validated by estimating the complex Ginzburg–Landau equation and turbulent channel flows with the friction Reynolds number $Re_{\tau }=186$, 547 and 934, where the results from the RBE are also included. When the measurements are temporally resolved, the estimation results of the two approaches are equivalent to each other, and both match well with the reference numerical results. For the temporally unresolved cases, the estimation errors from the GRBE are obviously lower than those from the RBE. After validation, the GRBE is applied to investigate the impacts of the abundance of the measured information, including the temporal information and sensor types, on the estimation accuracy. Compared with the mean square error (MSE) in the estimation with temporally resolved measurements, that with measurements at only one snapshot, i.e. without any temporal information, increases by approximately $15\,\%$. On the other hand, it can effectively improve the estimation accuracy by increasing the number of sensor types. With temporally resolved measurements, the relative MSE decreases by $12.3\,\%$ when the sensor types increase from $\lbrace \tau _u \rbrace$ to $\lbrace \tau _u,\tau _w,p \rbrace$, where $\tau_u$, $\tau_w$ and p are the streamwise shear stress, spanwise shear stress and pressure at the wall. Finally, several existing forcing models are incorporated into the GRBE to investigate their performance in the linear estimation of flow state. The wall-distance-dependent model (W-model) results match well with the optimal linear estimations when the measurements are temporally unresolved. Meanwhile, with the increase of temporal information of the measurement, the estimation errors from the tested W-model and the scale-dependent model ($\lambda$-model) both increase, which contradicts the tendency observed in the optimal linear GRBE estimation results. Such a phenomenon highlights the importance of proper modelling of the forcing in the temporal domain for the accuracy of flow state estimation.

Liquid bridges are formed when a flowing liquid interacts with multiple parallel fibres, as relevant to heat and mass transfer applications that utilize flow down fibre arrays. We perform a comprehensive experimental study of flowing liquid bridges between two vertical fibres whose spacing is controlled dynamically in our experimental apparatus. The bridge patterns exhibit a regular periodic spacing typical of absolute instability for low flow rates, but become spatially inhomogeneous above a critical flow rate where the base flow is convectively unstable. The shapes of individual bridges and their associated dynamics are measured, as they depend upon the liquid properties, and fibre geometry/spacing. The bridge length scales similarly to static bridges between parallel fibres. The bridge dynamics exhibits a dependence on viscosity and scale with the impedance. A simple energy balance is used to derive a scaling relationship for the bridge velocity that captures the general trend of our experimental data. Finally, we demonstrate that these scalings similarly apply when the fibres are dynamically separated or brought together.

The gain in efficiency of the receptivity of jets to acoustic disturbances as the nozzle lip is thicker is investigated using numerical simulations. For that, axisymmetric acoustic pulses are introduced in jets with Blasius laminar boundary-layer profiles at Mach numbers $M=0.4$, 0.6, 0.9 and 1.3 for nozzle-lip thicknesses between 1 % and 93 % of the nozzle radius. They are located on the jet axis or outside the jet with incidence angles $\varphi$ between $5^{\circ }$ and $90^{\circ }$ with respect to the downstream direction. Instability waves develop in the jet shear layer after the acoustic disturbances hit the nozzle. In all cases except for $\varphi \geq 75^{\circ }$, their amplitudes and hence the efficiency of the jet receptivity to the disturbances increase with the nozzle-lip thickness. The gains in efficiency are greater for a pulse inside the jet, generating upstream-travelling pressure waves resembling guided jet waves, than for a pulse outside the jet, producing free-stream sound waves. In the second case, the gains are significant for $\varphi =5^{\circ }$ and decrease with the incidence angle, especially for $\varphi >30^{\circ }$. Moreover, the gains are stronger for a higher Mach number, and roughly double between $M=0.4$ and $1.3$, thus reaching, for a pulse inside the jet, values close to 6 between the thinnest and thickest lips. Finally, according to additional simulations for $M=0.9$, the gains in receptivity efficiency do not change appreciably when different azimuthal mode numbers of the acoustic disturbances, widths of the pulse and shapes of the boundary-layer profile are considered.

A moving static pressure distribution is commonly used to simulate a travelling ship. However, the ship movement changes the fluid velocity around the hull, inducing pressures on the hull surface that are no longer equal to the static pressure. Therefore, we introduce a dynamic pressure correction strategy, which can accurately simulate the impact of the ship movement on the hull-surface pressure and preserve the desired hull shape under both stationary and transient conditions. The strategy is applied to a high-order spectral model and used to investigate ship-induced waves and wave resistance over a both flat and variable topography. We explore various parameters in our study, including the average water depth to ship draft ratio ($h_0/d$), the channel width to ship width ratio ($W/B$), the Froude number ($Fr_0=U/\sqrt {gh_0}$) and variations in bathymetric slope. Compared with experiments on a flat bottom, the numerical results with dynamic correction show better accuracy in the simulation of ship-induced waves and wave resistance than those obtained using a static pressure distribution. The correlation coefficient for wake waves between the numerical and experimental results is improved by approximately 0.25 with the dynamic correction strategy. The amplitude and wavelength of ship-induced mini-tsunamis over a variable topography are found to be reduced when employing a dynamic correction compared with a static pressure distribution, and this effect becomes more pronounced with higher Froude number. The static pressure approach is shown to allow large deformations of the desired hull shape and changes in ship volume which are responsible for the different wave patterns from the two approaches.

The impact of shoaling on linear water waves is well known, but it has only been recently found to significantly amplify both the intensity and frequency of rogue waves in nonlinear irregular wave trains atop coastal shoals. At least qualitatively, this effect has been partially attributed to the ‘rapid’ nature of the shoaling process, i.e. shoaling occurs over a distance far shorter than that required for waves to modulate themselves and adapt to the reduced water depth. Through a theoretical model and highly accurate nonlinear simulations, we disentangle the respective effects of the length and angle of a shoal's slope. We investigate the effects of the shoaling process rapidness on the evolution of key statistical and spectral sea-state parameters. We let the wave field evolve over a slope with constant angle in all cases while we vary the slope length. Our results indicate that the non-equilibrium dynamics is not affected by the slope length, because further extending the slope length does not influence the magnitude of the statistical and spectral measures as long as the non-equilibrium dynamics dominates the wave evolution. Thus, the shoaling effect on rogue waves is deduced to be mainly driven by the slope magnitude rather than the slope length.

The spreading of large viscous drops of density-matched suspensions of non-Brownian spheres on a smooth solid surface is investigated experimentally at the global drop scale. The focus is on dense suspensions with a solid volume fraction equal to or greater than $40\,\%$, and for drops larger than the capillary length, i.e. for which the spreading is governed by the balance of gravitational and viscous forces. Our findings indicate that all liquids exhibit a power-law behaviour typical of gravity-driven dynamics, albeit with an effective suspension viscosity that is smaller than the bulk value. When the height of the drop is of the order of the particle size, the power law breaks down as the particles freeze while the contact line continues to advance.

The dynamic behaviour of granular media can be observed widely in nature and in many industrial processes. Yet, the modelling of such media remains challenging as they may act with solid-like and fluid-like properties depending on the rate of the flow and can display a varying apparent friction, cohesion and compressibility. Over the last two decades, the $\mu (I)$-rheology has become well established for modelling granular liquids in a fluid mechanics framework where the apparent friction $\mu$ depends on the inertial number $I$. In the geo-mechanics community, modelling the deformation of granular solids typically relies on concepts from critical state soil mechanics. Along the lines of recent attempts to combine critical state and the $\mu (I)$-rheology, we develop a continuum model based on modified cam-clay in an elastoplastic framework which recovers the $\mu (I)$-rheology under flow. This model permits a treatment of plastic compressibility in systems with or without cohesion, where the cohesion is assumed to be the result of persistent inter-granular attractive forces. Implemented in a two- and three-dimensional material point method, it allows for the trivial treatment of the free surface. The proposed model approximately reproduces analytical solutions of steady-state cohesionless flow and is further compared with previous cohesive and cohesionless experiments. In particular, satisfactory agreements with several experiments of granular collapse are demonstrated, albeit with shear bands which can affect the smoothness of the surface. Finally, the model is able to qualitatively reproduce the multiple steady-state solutions of granular flow recently observed in experiments of flow over obstacles.

High-fidelity simulations are conducted to investigate the turbulent boundary layers around a finite-span NACA0012 wing with a rounded wing-tip geometry at a chord-based Reynolds number of $Re_c=200\,000$ and at various angles of attack up to $10^\circ$. The study aims to discern the differences between the boundary layers on the finite-span wing and those on infinite-span wings at equivalent angles of attack. The finite-span boundary layers exhibit: (i) an altered streamwise and a non-zero spanwise pressure gradient as a result of the variable downwash induced by the wing-tip vortices (an inviscid effect typical of finite-span wings); (ii) differences in the flow history at different wall-normal distances, caused by the variable flow angle in the wall-normal direction (due to constant pressure gradients and variable momentum normal to the wall); (iii) laminar flow entrainment into the turbulent boundary layers near the wing tip (due to a laminar–turbulent interface); and (iv) variations in boundary layer thickness across the span, attributed to the variable wall-normal velocity in that direction (a primarily inviscid effect). These physical effects are then used to explain the differences in the Reynolds stress profiles and other boundary layer quantities, including the reduced near-wall peak of the streamwise Reynolds stress and the elevated Reynolds stress levels near the boundary layer edge, both observed in the finite-span wings. Other aspects of the flow, such as the downstream development of wing-tip vortices and their interactions with the surrounding flow, are reserved for future investigations.

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

A macroscopic model for perfect-slip flow in porous media is derived in this work, starting from the pore-scale flow problem and making use of an upscaling technique based on the adjoint method and Green's formula. It is shown that the averaged momentum equation has a Darcy form in which the permeability tensor, $\boldsymbol{\mathsf{K}}_{ps}$, is obtained from an associated adjoint (closure) problem that is to be solved on a (periodic) unit cell representative of the structure. Similarly to the classical permeability tensor, $\boldsymbol{\mathsf{K}}$, in the no-slip regime, $\boldsymbol{\mathsf{K}}_{ps}$ is intrinsic to the porous medium under consideration and is shown to be symmetric and positive. Integral relationships between $\boldsymbol{\mathsf{K}}_{ps}$, the partial-slip flow permeability tensor, $\boldsymbol{\mathsf{K}}_{s}$, and $\boldsymbol{\mathsf{K}}$ are derived. Numerical simulations carried out on two-dimensional model porous structures, together with an approximate analytical solution and an empirical correlation for a particular configuration, confirm the validity of the macroscopic model and the relationship between $\boldsymbol{\mathsf{K}}_{ps}$ and $\boldsymbol{\mathsf{K}}$.