Iceberg melting is a critical factor for climate change. However, the shape of an iceberg is an often neglected aspect of its melting process. Our study investigates the influence of different ice shapes and ambient flow velocities on melt rates by conducting direct numerical simulations of a simplified system of bluff body flow. Our study focuses on the ellipsoidal shape, with the aspect ratio as the control parameter. We found the shape plays a crucial role in the melting process, resulting in significant variations in the melt rate between different shapes. Without flow, the optimal shape for a minimal melt rate is the disk (two-dimensional) or sphere (three-dimensional), due to the minimal surface area. However, as the ambient flow velocity increases, the optimal shape changes with the aspect ratio. We find that ice with an elliptical shape (when the long axis is aligned with the flow direction) can melt up to 10 % slower than a circular shape when exposed to flowing water. Following the approach considered by Huang et al. (J. Fluid Mech., vol. 765, 2015, R3) for dissolving bodies, we provide a quantitative theoretical explanation for this optimal shape, based on the combined contributions from both surface-area effects and convective-heat-transfer effects. Our findings provide insight into the interplay between phase transitions and ambient flows, contributing to our understanding of the iceberg melting process and highlighting the need to consider the aspect-ratio effect in estimates of iceberg melt rates.

Simultaneous time-resolved measurements of wall deformation and the 3-D velocity field in boundary layers over a compliant surface are performed by integrating Mach Zehnder interferometry with tomographic particle tracking velocimetry. The pressure is calculated by spatially integrating the material acceleration. Combining data obtained from several references, trends of the deformation r.m.s. scaled by the compliant wall thickness collapse when plotted vs pressure fluctuations scaled by the material shear modulus. For the present data, at all Reynolds numbers, the deformation waves travel at 53% of the free-stream velocity and have a preferred wavelength of three times the thickness. The latter is consistent with theoretical models. Adopting insight derived from atmospheric wind–wave interactions, the pressure–deformation correlations peak at or slightly above the ‘critical layer’, where the mean flow speed is equal to the surface wave speed. This layer is located within the log layer, and when expressed using inner variables, increases in elevation with increasing Reynolds number. For the entire region below the critical layer, wavenumber–frequency spectra of pressure and vertical velocity fluctuations indicate that the turbulence is phase locked and travels with the deformation, even for deformation amplitudes much smaller than a wall unit. In contrast, above the critical layer, the turbulence is advected at the local mean streamwise velocity, and its correlation with the deformation decays rapidly. These findings indicate that the height of the zone dominated by flow-deformation interactions is determined by the surface wave speed, and its variations are caused by deformation-induced modifications to the mean velocity profile.

The dynamics of soft porous media involves complex interactions between fluid flow and elasticity. The recent paper by Fiori et al. (J. Fluid Mech., vol. 974, 2023, A2) highlights phenomena relating to the periodic loading of such poro-elastic media, including hysteresis and the localisation of deformation at high frequencies. These effects could result in rectification and steady streaming in many important applications.

In magnetoconvection, the flow of an electromagnetically conductive fluid is driven by a combination of buoyancy forces, which create the fluid motion due to thermal expansion and contraction, and Lorentz forces, which distort the convective flow structure in the presence of a magnetic field. The differences in the global flow structures in the buoyancy-dominated and Lorentz-force-dominated regimes lead to different heat transport properties in these regimes, reflected in distinct dimensionless scaling relations of the global heat flux (Nusselt number $Nu$) versus the strength of buoyancy (Rayleigh number $Ra$) and electromagnetic forces (Hartmann number $Ha$). Here, we propose a theoretical model for the transition between these two regimes for the case of a static vertical magnetic field applied across a convective fluid layer confined between two isothermal, a lower warmer and an upper colder, horizontal surfaces. The model suggests that the scaling exponents $\gamma$ in the buoyancy-dominated regime, $Nu\sim Ra ^\gamma$, and $\xi$ in the Lorentz-force-dominated regime, $Nu\sim (Ha^{-2}Ra)^\xi$, are related as $\xi =\gamma /(1-2\gamma )$, and the onset of the transition scales with $Ha^{-1/\gamma }Ra$. These theoretical results are supported by our direct numerical simulations for $10\leq Ha\leq 2000$, Prandtl number $Pr=0.025$ and $Ra$ up to $10^9$ and data from the literature.

We investigate and compare various types of acoustic trapped modes (TMs) in resonator–waveguide systems. The goal is to understand the commonality and difference between the mechanisms of common (symmetry protected, invisibility protected and symmetry–periodicity protected) and accidental TMs, occurring continuously and discretely in the resonator length–frequency two-parameter space. The latter type cannot yet be explained via an operator decomposition. Here, all TMs are explained in the same way by analysing why and how the propagating-wave loops in the eigenfunctions can satisfy the eigenmode condition (loop magnitude and phase constraints for closure) and the wave-trapping condition (loop zero-radiation condition) simultaneously. Firstly, the unified analysis reveals the commonality that one or multiple coupled propagating-wave loops satisfy TM conditions, and the difference. In common TMs, the loop zero radiation is independent of the single loop phase constraint that selects the TM frequency as a continuous function of resonator length. On the other hand, loop zero radiation in accidental TMs depends on the loop phase constraints and there are two phase constraints. Only at the crossing points of the two phase constraints can zero-radiation loops be ensured. Secondly, in contrast to previous studies, it suggests that modal degeneracy, avoided crossing and resonance-width bifurcation are not the mechanisms of accidental TMs.

The understanding of the entrainment mechanism of synthetic jets can help optimise the synthetic jet actuators in engineering applications. It is generally believed that vortex rings or strong velocity fluctuations in the near field of the synthetic jet are responsible for its enhanced entrainment. However, in recent years, it has been found that the enhanced entrainment of the synthetic jet may be caused by the instability or the vortex ring breakdown in the transition region. To shed new light on this issue, synthetic jets with different Reynolds numbers and dimensionless stroke lengths are investigated with time-resolved two-dimensional particle image velocimetry. Based on the analyses of velocity triple-decomposition, Fourier mode decomposition and phase-averaged $\lambda _{ci}D/U_0$ field, the streamwise positions of the vortex ring breakdown are determined for the synthetic jets, and the entrainment coefficient can be divided into three components, i.e. the coherent turbulent kinetic energy production, the random turbulent kinetic energy production and the shape of the velocity profile. It is found that the entrainment coefficient is dominated by the component related to the random turbulent kinetic energy production, and reaches its peak value at the position of vortex ring breakdown. The results obtained in different cases show a strong correlation between vortex ring breakdown and entrainment enhancement. From the perspective of instantaneous snapshot, the mechanism of vortex ring breakdown enhanced entrainment is revealed, that is, vortex ring breakdown enhanced the small-scale vortex near the turbulent/non-turbulent interface, resulting in an increase of enstrophy production, and thus enhanced local entrainment.

Spontaneous motion due to symmetry breaking has been predicted theoretically for both active droplets and isotropically active particles in an unbounded fluid domain, provided that their intrinsic Péclet number $Pe$ exceeds a critical value. However, due to their inherently small $Pe$, this phenomenon has yet to be observed experimentally for active particles. In this paper, we demonstrate theoretically that spontaneous motion for an active spherical particle closely fitting in a cylindrical channel is possible at arbitrarily small $Pe$. Scaling arguments in the limit where the dimensionless clearance is $\epsilon \ll 1$ reveal that when $Pe=O(\epsilon ^{1/2})$, the confined particle reaches speeds comparable to those achieved in an unbounded fluid at moderate (supercritical) $Pe$ values. We use matched asymptotic expansions in that distinguished limit, where the fluid domain decomposes into several asymptotic regions: a gap region, where the lubrication approximation applies; particle-scale regions, where the concentration is uniform; and far-field regions, where solute transport is one-dimensional. We derive an asymptotic formula for the particle speed, which is a monotonically decreasing function of $\overline {Pe}=Pe/\epsilon ^{1/2}$ and approaches a finite limit as $\overline {Pe}\searrow 0$. Our results could pave the way for experimental realisations of symmetry-breaking spontaneous motion in active particles.

Direct numerical simulations are conducted for temporally evolving stratified wake flows at Reynolds numbers from $10\,000$ to $50\,000$ and Froude numbers from $2$ to 50. Unlike previous studies that obtained statistics from a single realization, we take ensemble averages among 80–100 realizations. Our analysis shows that data from one realization incur large convergence errors. These errors reduce quickly as the number of statistical samples increases, with the benefit of ensemble average diminishing beyond 40–60 realizations. The data with ensemble average allow us to test the previously established scalings and arrive at new scaling estimates. Specifically, the data do not support power-law scaling in the centreline velocity deficit $U_0$ beyond the near wake. Its decay rate increases continuously from 0.1 at the onset of the non-equilibrium regime until the end of our calculations without reaching any asymptote. Additionally, while no power-law scalings could be found in the wake width ($L_H$) and wake height ($L_V$) in the late wake, $L_H\sim (Nt)^{1/3}$ is a good working approximation of the wake's horizontal size, where $N$ is buoyancy frequency and $t$ is time. Besides the low-order statistics, we also report the transverse integrated terms and the vertically integrated terms in the turbulent kinetic energy budget equation as a function of the vertical and transverse coordinates. The data indicate that there are two peaks in the vertically integrated production and transport terms, and one peak when the two terms are integrated horizontally.

Experimental and theoretical studies on millimetre-sized droplets suggest that at low Reynolds number the difference between the drag force on a circulating water droplet and that on a rigid sphere is very small (less than 1 %) (LeClair et al., J. Atmos. Sci., vol. 29, 1972, pp. 728–740). While the drag force on a spherical liquid droplet at high viscosity ratios (of the liquid to the gas), is approximately the same as that on a rigid sphere of the same size, the other quantities of interest (e.g. the temperature) in the case of a rarefied gas flow over a liquid droplet differ from the same quantities in the case of a rarefied gas flow over a rigid sphere. The goal of this article is to study the effects of internal motion within a spherical microdroplet/nanodroplet – such that its diameter is comparable to the mean free path of the surrounding gas – on the drag force and its overall dynamics. To this end, the problem of a slow rarefied gas flowing over an incompressible liquid droplet is investigated analytically by considering the internal motion of the liquid inside the droplet and also by accounting for kinetic effects in the gas. Detailed results for different values of the Knudsen number, the ratio of the thermal conductivities and the ratio of viscosities are presented for the pressure and temperature profiles inside and outside the liquid droplet. The results for the drag force obtained in the present work are in good agreement with the theoretical and experimental results existing in the literature.

We consider inertial waves propagating in a fluid contained in a non-axisymmetric three-dimensional rotating cavity. We focus on the particular case of a fluid enclosed inside a truncated cone or frustum, which is the volume that lies between two horizontal parallel planes cutting an upright cone. While this geometry has been studied in the past, we generalise it by breaking its axisymmetry and consider the case of a truncated elliptic cone for which the horizontal sections are elliptic instead of circular. The problem is first tackled using ray tracing, where local wave packets are geometrically propagated and reflected within the closed volume without attenuation. We complement these results with a local asymptotic analysis and numerical simulations of the original linear viscous problem. We show that the attractors, well known in two dimensional or axisymmetric domains, can be trapped in a particular plane in three dimensions provided that the axisymmetry of the domain is broken. Contrary to previous examples of attractors in three-dimensional domains, all rays converge towards the same limit cycle regardless of initial conditions, and it is localised in the bulk of the fluid.

Recent microfluidic experiments have evidenced complex spatio-temporal fluctuations in low-Reynolds-number flows of polymer solutions through lattices of obstacles. However, understanding the nonlinear physics of such systems remains a challenge. Here, we use high performance simulations to study viscoelastic flows through a hexagonal lattice of cylindrical obstacles. We find that structures of localized polymer stress – in particular birefringent strands – control the stability and the dynamics. We first show that, at steady state, strands act as a web of sticky flow barriers that induce channelization, multistability and hysteresis. We then demonstrate that a spontaneous destabilization of the strands drives the transition to unsteady flow with regimes of self-sustained oscillations, travelling waves and strand pulsations. We further show that these pulsations, which result from the destabilization of envelope patterns of stress with strands wrapped around multiple obstacles, are integral to the transition towards elastic turbulence in our two-dimensional simulations. Our study provides a new perspective on the role of birefringent strands and a framework for understanding experimental observations. We anticipate that it is an important step towards unifying existing interpretations of the nonlinear physics of viscoelastic flows through complex structures.

Very-large-scale motions are commonly observed in moderate- and high-Reynolds-number wall turbulence, constituting a considerable portion of the Reynolds stress and skin friction. This study aims to investigate the behaviour of these motions in high-speed and high-Reynolds-number turbulent boundary layers at varying Mach numbers. With the aid of high-precision numerical simulations, numerical experiments and theoretical analysis, it is demonstrated that the very-large-scale motions are weakened in high-Mach-number turbulence at the same friction Reynolds numbers, leading to the reduction in turbulent kinetic energy in the outer region. Conversely, the lower wall temperature enhances the very-large-scale motions but shortens the scale separation between the structures in the near-wall and outer regions.

A theoretical and computational study analysing the initiation of yield-stress fluid percolation in porous media is presented. Yield-stress fluid flows through porous media are complicated due to the nonlinear rheological behaviour of this type of fluid, rendering the conventional Darcy type approach invalid. A critical pressure gradient must be exceeded to commence the flow of a yield-stress fluid in a porous medium. As the first step in generalising the Darcy law for yield-stress fluids, a universal scale based on the variational formulation of the energy equation is derived for the critical pressure gradient which reduces to the purely geometrical feature of the porous media. The presented scaling is then validated by both exhaustive numerical simulations (using an adaptive finite element approach based on the augmented Lagrangian method), and also the previously published data. The considered porous media are constructed by randomised obstacles with various topologies; namely square, circular and alternatively polygonal obstacles which are mimicked based on Voronoi tessellation of circular cases. Moreover, computations for the bidispersed obstacle cases are performed which further demonstrate the validity of the proposed universal scaling.

Wall modelling in large-eddy simulation (LES) is of high importance to allow scale resolving simulations of industrial applications. Numerous models were developed and validated for incompressible flows, including a simple quasi-analytical model based on Reichardt's formula that approximates the law of the wall. In this paper, a scaling is proposed to generalize this wall model to highly compressible flows. First, the results of wall-resolved LES (wrLES) of adiabatic compressible turbulent channel flows at $Re_\tau = 1000$ and at centreline Mach numbers of $M_c= 0.76$ and $1.5$ are presented. Then, three potential scalings of the incompressible wall model are proposed, and their a priori performance is evaluated : (i) the Howarth–Stewartson scaling, (ii) an improved Van Driest scaling and (iii) a new scaling obtained from a blending of those two. The results of wall-modelled LES (wmLES) of compressible channel flows using these three models are compared with the reference wrLES data, showing the superior accuracy of the hybrid scaling. The consistency of the new wall model at low Mach numbers is also verified by comparing the results of a wmLES at $M_c= 0.25$ with those of reference incompressible DNS data at $Re_\tau = 1000$ and $5200$. Finally, the proposed wall model is also applied to a turbulent channel flow at $M_c=1.5$ and $Re_\tau =5200$.

A data-driven implementation of a quasi-linear approximation is presented, extending a minimal quasi-linear approximation (MQLA) (Hwang & Ekchardt, J. Fluid Mech., vol. 894, 2020, p. A23) to incorporate non-zero streamwise Fourier modes. A data-based approach is proposed, matching the two-dimensional wavenumber spectra for a fixed spanwise wavenumber between a direct numerical simulation (DNS) (Lee & Moser, J. Fluid Mech., vol. 774, 2015, pp. 395–415) and that generated by the eddy viscosity enhanced linearised Navier–Stokes equations at $Re_\tau \approx 5200$, where $Re_\tau$ is the friction Reynolds number. Leveraging the self-similar nature of the energy-containing part in the DNS velocity spectra, a universal self-similar streamwise wavenumber weight is determined for the linearised fluctuation equations at $Re_\tau \simeq ~5200$. The data-driven quasi-linear approximation (DQLA) provides noteworthy enhancements in the wall-normal and spanwise turbulence intensity profiles. It exhibits a qualitatively similar structure in the spanwise wavenumber velocity spectra compared with the MQLA. Additionally, the DQLA offers extra statistical outputs in the streamwise wavenumber coordinates, enabling a comprehensive global analysis of this modelling approach. By comparing the DQLA results with DNS results, the limitations of the presented framework are discussed, mainly pertaining to the lack of the streak instability (or transient growth) mechanism and energy cascade from the linearised model. The DQLA is subsequently employed over a range of Reynolds numbers up to $Re_\tau = 10^5$. Overall, the turbulence statistics and spectra produced by the DQLA scale consistently with the available DNS and experimental data, with the Townsend–Perry constants displaying a mild Reynolds dependence (Hwang, Hutchins & Marusic, J. Fluid Mech., vol. 933, 2022, p. A8). The scaling behaviour of the turbulence intensity profiles deviates away from the classic $\ln (Re_\tau )$ scaling, following the inverse centreline velocity scaling for the higher Reynolds numbers.

Recent advances in manufacturing techniques are providing porous media with both high permeability, necessary for effective passive flow control, and high structural strength, essential for engineering applications. We therefore examine the predictive accuracy of the standard Darcy–Forchheimer (DF) law, which is often used to model porous media flows, for inclusion Reynolds numbers (Re) ranging from the low linear regime to the high nonlinear regime where unsteady effects such as vortex shedding become evident. We consider two different inclusion shapes, square and circular, and three different arrangements of the inclusions – inline, staggered and random. The numerical simulations show that the DF law performs well for low-Re flows, irrespective of the inclusion configuration. For intermediate/high-Re flows, the DF law is adequate only when the arrangement is highly random. However, for the regularly arranged topologies or less random geometries at intermediate/high-Re flows, the DF-law performance diminishes significantly due to flow sheltering and redirection (‘inertial flow deflection’) effects that arise from flow inertia, separation and vortex shedding in the wake of the inclusions. It is shown that the standard DF law, in which the nonlinear permeability tensor is independent of orientation, does not capture such effects. We modified the DF law to capture flow redirection effects by allowing the Forchheimer permeability tensor to depend on the flow orientation with respect to the principal geometrical directions of the porous geometry, and examined this extended DF law for these flows.

A fundamental open problem in the mechanics of granular media is the determination of the stress in the static state. It is known that the static stress depends strongly on how the grain assembly is created and the nature of confining boundaries. Non-trivial spatial variations have been observed even in simple geometries, posing long-standing challenges to continuum modelling. In this paper, we create gravity-deposited grain packings computationally and devise a method to visualise the paths of load transmission, which we call force lines. We show that the force lines reflect the flow during deposition, thereby encoding preparation history. We then show that the force lines coincide with ensemble averaged biased random walks in the particle contact network; this identification yields a closure relation for the stress, which together with the static momentum balances fully determines the stress field. The model makes accurate predictions for the stress in piles and silos, even for unusual deposition methods, thereby showing promise for more general scenarios.

In an effort to capture the continuous hydraulic jump and flow structure for a jet impinging on a disk, we recently proposed a composite mean-field thin-film approach consisting of subdividing the flow domain into three distinct connected regions of increasing gravity strength (Wang et al., J. Fluid Mech., vol. 966, 2023, A15). In the present study, we further validate our approach, and examine the characteristics and structure of the circular jump and recirculation. The influence of the disk radius is found to be significant, especially in the subcritical region. Below a disk radius, the jump transits from type Ia to type 0 after the recirculation zone has faded. The supercritical flow and jump location are insensitive to the disk size, but the jump length and height as well as the vortex size are strongly affected, all decreasing with decreasing disk radius, exhibiting a maximum with the flow rate for a small disk. The jump is relatively steep with a strong recirculation zone for a high obstacle at the disk edge. Comparison against the Navier–Stokes solution of Askarizadeh et al. (Phys. Rev. Fluids, vol. 4, 2019, 114002; Intl J. Heat Mass Transfer, vol. 146, 2020, 118823) for the weak and intermediate surface tension suggests that the surface tension effect is unimportant for a high obstacle for a jump of type 0 or type Ia. The film thickness at the disk edge for a freely draining film is found to comprise, in addition to a static component (capillary length), a dynamic component: ${h_\infty }\sim {(Fr/{r_\infty })^{2/3}}$ that we establish by minimizing the Gibbs free energy at the disk edge, and, equivalently, is also the consequence of the flow becoming supercritical near the edge. By assuming negligible film slope and curvature at the leading edge of the jump and maximum height at the trailing edge, we show that the jump length is related to the jump radius as ${L_J}\sim Re{(F{r^2}/{r_J}^5)^{1/3}}$. The vortex length follows the same behaviour. The energy loss and conjugate depth ratio exhibit a maximum with the flow rate, which we show to originate from the descending and ascending branches of the supercritical film thickness. The presence of the jump is not necessarily commensurate with that of a recirculation; the existence of the vortex closely depends on the upstream curvature and steepness of the jump. The surface separating the regions of existence/non-existence of the recirculation is given by the universal relation $R{e^{10/3}}F{r^2} = 9r_\infty ^9/50$. The jump can be washed off the edge of the disk, particularly at low viscosity and small disk size. The flow in the supercritical region remains insensitive to the change in gravity level and disk size but is greatly affected by viscosity.

Field observations of debris flows often show that a deep dry granular front is followed by a progressively thinner and increasingly watery tail. These features have been captured in recent laboratory flume experiments (Taylor-Noonan et al., J. Geophys. Res.: Earth Surf., vol. 127, 2022, e2022JF006622). In these experiments different initial release volumes were used to investigate the dynamics of an undersaturated monodisperse grain–water mixture as it flowed downslope onto a horizontal run-out pad. Corresponding dry granular flows, with the same particle release volumes, were also studied to show the effect of the interstitial fluid. The inclusion of water makes debris flows much more mobile than equivalent volumes of dry grains. In the wet flows, the formation of a dry front is crucially dependent on the heterogeneous vertical structure of the flow and the velocity shear. These effects are included in the depth-averaged theory of Meng et al. (J. Fluid Mech., vol. 943, 2022, A19), which is used in this paper to quantitatively simulate both the wet and dry experimental flows using a high-resolution shock-capturing scheme. The results show that velocity shear causes dry grains (located near the free surface) to migrate forwards to create a dry front. The front is more resistant to motion than the more watery material behind, which reduces the overall computed run-out distance compared with debris-flow models that assume plug flow and develop only small dry snouts. Velocity shear also implies that there is a net transport of water to the back of the flow. This creates a thin oversaturated tail that is unstable to roll waves in agreement with experimental observations.

Plane unsteady potential flows of an ideal incompressible fluid with a free boundary are considered in the absence of external forces and surface tension. Examples of exact solutions in situations where the entire boundary of the domain occupied by the fluid is completely free are constructed. There may be polar singularities of the complex velocity function inside the fluid, which corresponds to the presence of a source or a sink there.