Depth-averaged systems of equations describing the motion of fluid–sediment mixtures have been widely adopted by scientists in pursuit of models that can predict the paths of dangerous overland flows of debris. As models have become increasingly sophisticated, many have been developed from a multi-phase perspective in which separate, but mutually coupled sets of equations govern the evolution of different components of the mixture. However, this creates the opportunity for the existence of pathological instabilities stemming from resonant interactions between the phases. With reference to the most popular approaches, analyses of two- and three-phase models are performed, which demonstrate that they are more often than not ill posed as initial-value problems over physically relevant parameter regimes – an issue which renders them unsuitable for scientific applications. Additionally, a general framework for detecting ill posedness in models with any number of phases is developed. This is used to show that small diffusive terms in the equations for momentum transport, which are sometimes neglected, can reliably eliminate this issue. Conditions are derived for the regularisation of models in this way, but they are typically not met by multi-phase models that feature diffusive terms.

We study experimentally the starting vortices shed by airfoils accelerating uniformly from rest in superfluid helium-4 (He II). The vortices behave apparently as if they were moving in a classical Newtonian fluid, such as air or water. Specifically, the starting vortex positions obtained from the experimental data are found to be very close to those computed numerically in a Newtonian fluid, at sufficiently small times, when self-similar behaviour is expected to occur, and for Reynolds numbers ranging between approximately $5 \times 10^2$ and $5 \times 10^5$. The result indicates neatly that turbulent flows of He II can be very similar to classical flows of Newtonian fluids, when thermal effects can be neglected and at sufficiently large flow scales, i.e. the study demonstrates that He II could also be employed to study classical Newtonian flows.

Eddies within the meso/submeso-scale range are prevalent throughout the Arctic Ocean, playing a pivotal role in regulating the freshwater budget, heat transfer and sea ice transport. While observations have suggested a strong connection between the dynamics of sea ice and the underlying turbulent flows, quantifying this relationship remains an ambitious task due to the challenges of acquiring concurrent sea ice and ocean measurements. Recently, an innovative study using a unique algorithm to track sea ice floes showed that ice floes can be used as vorticity-meters of the ocean. Here, we present a numerical and analytical evaluation of this result by estimating the kinematic link between free-drifting ice floes and underlying ocean eddies using idealised vortex models. These analyses are expanded to explore local eddies in quasi-geostrophic turbulence, providing a more realistic representation of eddies in the Arctic Ocean. We find that in both flow fields, the relationship between floe rotation rates and ocean vorticity depends on the relative size of the ice floe to the eddy. As the floe size approaches and exceeds the eddy size, the floe rotation rates depart from half of the ocean vorticity. Finally, the effects of ice floe thickness, atmospheric winds and floe collisions on floe rotations are investigated. The derived relations and floe statistics set the foundation for leveraging remote sensing observations of floe motions to characterise eddy vorticity at small to moderate scales. This innovative approach opens new possibilities for quantifying Arctic Ocean eddy characteristics, providing valuable inputs for more accurate climate projections.

We study buoyant miscible injections of dense viscoplastic fluids into lighter Newtonian fluids in inclined closed-end pipes, at the high-Péclet-number regime. We integrate experiments involving camera imaging and ultrasound Doppler velocimetry, and computational fluid dynamics simulations, to provide a detailed analysis of interfacial dynamics, flow phases/regimes, velocity field, yielded and unyielded zones, and interfacial arrest mechanisms. The flow dynamics is governed by Reynolds ($Re$), Froude ($Fr$) and Bingham ($B$) numbers, the viscosity ratio ($M$), inclination angle ($\beta$), or their combinations, such as $\chi \equiv 2Re/Fr^2$. As the interface evolves, our results reveal a transition from an inertial-dominated phase, characterised by linear front advancement at the injection velocity, to a viscoplastic-dominated phase, marked by deceleration and eventual interfacial arrest governed by the yield stress. The critical transition length between these phases $(\mathcal{L} \approx 1.26 Fr^{0.14})$ is determined by a balance between inertial and buoyant stresses. Experimental findings confirm buoyancy-driven slumping in our flows, consistent with the theoretical yield number criterion ($Y \equiv B/\chi$), with maximum interfacial arrest lengths scaling as $L_s \sim 1/Y$. These results also classify arrested and unhalted interfacial flow regimes on a plane involving ${\chi \cos (\beta )}/{B}$ and $Y$. Furthermore, we demonstrate that the interfacial arrest mechanism arises from interactions between buoyancy, rheology and geometry, as diminishing shear stresses promote unyielded zone expansion near the interface, progressively encompassing the viscoplastic layer and halting flow when stresses fall below the yield stress.

We consider the conceptual two-layered oscillating tank of Inoue & Smyth (2009 J. Phys. Oceanogr. vol. 39, no. 5, pp. 1150–1166), which mimics the time-periodic parallel shear flow generated by low-frequency (e.g. semi-diurnal tides) and small-angle oscillations of the density interface. Such self-induced shear of an oscillating pycnocline may provide an alternate pathway to pycnocline turbulence and diapycnal mixing in addition to the turbulence and mixing driven by wind-induced shear of the surface mixed layer. We theoretically investigate shear instabilities arising in the inviscid two-layered oscillating tank configuration and show that the equation governing the evolution of linear perturbations on the density interface is a Schrödinger-type ordinary differential equation with a periodic potential. The necessary and sufficient stability condition is governed by a non-dimensional parameter $\beta$ resembling the inverse Richardson number; for two layers of equal thickness, instability arises when $\beta \,{\gt}\,1/4$. When this condition is satisfied, the flow is initially stable but finally tunnels into the unstable region after reaching the time marking the turning point. Once unstable, perturbations grow exponentially and reveal characteristics of Kelvin–Helmholtz (KH) instability. The modified Airy function method, which is an improved variant of the Wentzel–Kramers–Brillouin theory, is implemented to obtain a uniformly valid, composite approximate solution to the interface evolution. Next, we analyse the fully nonlinear stages of interface evolution by modifying the circulation evolution equation in the standard vortex blob method, which reveals that the interface rolls up into KH billows. Finally, we undertake real case studies of Lake Geneva and Chesapeake Bay to provide a physical perspective.

We investigate the energy transfer from the mean profile to velocity fluctuations in channel flow by calculating nonlinear optimal disturbances, i.e. the initial condition of a given finite energy that achieves the highest possible energy growth during a given fixed time horizon. It is found that for a large range of time horizons and initial disturbance energies, the nonlinear optimal exhibits streak spacing and amplitude consistent with direct numerical simulation (DNS) at least at ${Re}_\tau = 180$, which suggests that they isolate the relevant physical mechanisms that sustain turbulence. Moreover, the time horizon necessary for a nonlinear disturbance to outperform a linear optimal is consistent with previous DNS-based estimates using eddy turnover time, which offers a new perspective on how some turbulent time scales are determined.

We simulate thermal convection in a two-dimensional square box using the no-slip condition on all boundaries, and isothermal bottom and top walls, and adiabatic sidewalls. We choose 0.1 and 1 for the Prandtl number $Pr$ and vary the Rayleigh number $Ra$ between $10^6$ and $10^{12}$. We particularly study the temporal evolution of integral transport quantities towards their steady states. Perhaps not surprisingly, the velocity field evolves more slowly than the thermal field, and its steady state – which is nominal in the sense that large-amplitude low-frequency oscillations persist around plausible averages – is reached exponentially. We study these oscillation characteristics. The transient time for the velocity field to achieve its nominal steady state increases almost linearly with the Reynolds number. For large $Ra$, the Reynolds number itself scales almost as $Ra^{2/3}\, Pr^{-1}$, and the Nusselt number as $Ra^{2/7}$.

The wake merging of two side-by-side porous discs with varying disc spacing is investigated experimentally in a wind tunnel. Two disc designs used in the literature are employed: a non-uniform disc and a mesh disc. Hot-wire anemometry is utilised to acquire two spanwise profiles at 8 and 30 disc diameters downstream and along the centreline between the dual-disc configuration up to 40 diameters downstream. The spanwise Castaing parameter profiles confirm the appearance of rings of internal intermittency at the outermost parts of the wakes. These rings are the first feature to interact between the discs. After this point, the turbulence develops to a state whereby an inertial range is observable in the spectra. Farther downstream, the internal intermittency shows the classical features of homogeneous, isotropic turbulence. These events are repeatable and occur in the same order for both types of porous discs. This robustness allows us to develop a general map of the merging of the two wakes.

This paper presents an experimental and analytical investigation into the use of trailing edge slits for the reduction of aerofoil trailing edge noise. The noise reduction mechanism is shown to be fundamentally different from conventional trailing edge serrations, relying on destructive interference from highly compact and coherent sources generated at either ends of the slit. This novel approach is the first to exploit the coherence intrinsic to the boundary layer turbulence. Furthermore, the study demonstrates that trailing edge slits not only achieve superior noise reductions compared with sawtooth serrations of the same amplitude at certain conditions, but also offer frequency-tuning capability for noise reduction. Noise reduction is driven by the destructive interference between acoustic sources at the root and tip of the slit, which radiate with a phase difference determined by the difference in times taken for the boundary layer flow to convect between the root and tip. Maximum noise reductions occur at frequencies where the phase difference between these sources is $180^\circ$. The paper also presents a detailed parametric study into the variation in noise reductions due to the slit length, slit wavelength and slit root width. Additionally, a simple two-source analytic model is proposed to explain the observed results. Wind tunnel measurements of the unsteady flow field around the trailing edge slits are also presented, providing insights into the underlying flow physics.

In this study, changes in the mean flow of a compressible turbulent boundary layer spatially evolving from low to ‘moderate’ Reynolds numbers are examined. All discussions are based on literature data and a direct numerical simulation (DNS) of a supersonic boundary layer specifically designed to be effectively free of spurious inflow effects in the range $4000 \lessapprox Re_\theta \lessapprox 5000$, which enables discussion of sensitive properties such as the turbulent wake. Most noticeably, the DNS data show the formation of a distinct ‘bend’ in the friction coefficient distribution reflected in sudden deviation from established low-Reynolds-number correlations. As will be shown, the bend is related to the surprisingly abrupt saturation of the turbulent wake, marking the change from low- to moderate-Reynolds-number behaviour; in previous studies, this trend was potentially obscured by data scatter in experiments and/or insufficient domain length in DNS. Moreover, the influence of the wake saturation on the formation of the early logarithmic overlap layer is assessed, which, if fully developed, leads to the onset of high-Reynolds-number behaviour further downstream.

Modelling the nonlinear forcing is critical for linear models based on resolvent or input–output analyses. For compressible wall-bounded turbulence, little is known on what the real forcing looks like due to limited data, so the prediction agrees more qualitatively than quantitatively with direct numerical simulations (DNSs). Here, we present detailed forcing statistics of stochastic linear models, derived from elaborate DNS datasets for channel flows with bulk Mach number reaching 3. These statistics directly explain the success and failure of current models and provide guidance for further improvements. The benchmark linearised Navier–Stokes (LNS) and eLNS models are considered; the latter is assisted by eddy-viscosity-related terms. First, we prove the self-consistency of the models by using DNS-computed forcing as the input. Second, we present the spectral distributions of the forcing and its components. Third, we quantify the acoustic components, absent in incompressible cases, within the linear models. We reveal that the LNS forcing can exhibit relatively high coherence and low rank, very different from the modelled diagonal full-rank forcing. The eddy-viscosity-related term is not partial modelling of the LNS forcing; contrarily, the former is much larger than the latter, serving to disrupt the low-rank feature, enhance diagonal dominance and increase robustness across scales. The scales narrow in either horizontal direction are most susceptible to acoustic modes, while the others are little affected (${\lt}2\,\%$ in energy). Furthermore, the extended strong Reynolds analogy is assessed in predicting the density and temperature components.

In a combined experimental and numerical effort, we investigate the generation and reduction of airfoil tonal noise. The means of noise control are streak generators in the form of cylindrical roughness elements. These elements are placed periodically along the span of the airfoil at the mid-chord streamwise position. Experiments are performed for a wide range of Reynolds numbers and angles of attack in a companion work (Alva et al., AIAA Aviation Forum, 2023). In the present work, we concentrate on numerical investigations for a further investigation of selected cases. We have performed wall-resolved large-eddy simulations for a NACA 0012 airfoil at zero angle of attack and Mach 0.3. Two Reynolds numbers (${0.8\times 10^{5}}$ and ${1.0 \times 10^{5}}$) have been investigated, showing acoustic results consistent with experiments at the same Reynolds but lower Mach numbers. Roughness elements attenuate tones in the acoustic field and, for the higher Reynolds number, suppress them. Through Fourier decomposition and spectral proper orthogonal decomposition analysis of streamwise velocity data, dominating structures have been identified. Further, the coupling between the structures generated by the surface roughness and the instability modes (Kelvin–Helmholtz) of the shear layer has been identified through stability analysis, suggesting stabilisation mechanisms by which the sound generation by the airfoil is reduced by the roughness elements.

An analytical formulation is provided that describes the first two natural modes of the fluid–structure interaction of an incompressible current with a pitching and heaving flexible plate. The objective is twofold: first, to present a general derivation of analytical expressions for the lift, moment and the flexural moments exerted by an inviscid flow on a pitching and heaving plate whose deformation is general enough that the coupling of the flexural moments with the structural equations allows solving analytically the first two natural modes of the system; second, to analyse the propulsion performance of the foil when actuated near the first two natural frequencies. For the second purpose, one also needs the thrust force generated through the motion and the general deformation of the foil considered, which is analytically derived using the linearized vortex impulse theory, extending and systematizing previous works. The analytical expressions, once viscous effects are taken into consideration through nonlinear transverse damping and offset drag coefficients, are compared with small-amplitude available experimental data, discussing their limitations. It is found that low stiffness pitching and heaving are quite different, with a pitching flexible foil only generating thrust near the second resonant frequency, whereas heaving always generates thrust, with the maximum slightly below the second natural frequency. Maximum thrust for large stiffness pitching is around the first natural frequency. The maximum efficiency occurs at frequencies close to the first natural mode if the foil is sufficiently rigid, but it is not related to the natural frequencies as the rigidity decreases.

Periodic travelling waves at the free surface of an incompressible inviscid fluid in two dimensions under gravity are numerically computed for an arbitrary vorticity distribution. The fluid domain over one period is conformally mapped from a fixed rectangular one, where the governing equations along with the conformal mapping are solved using a finite-difference scheme. This approach accommodates internal stagnation points, critical layers and overhanging profiles, thereby overcoming limitations of previous studies. The numerical method is validated through comparisons with known solutions for zero and constant vorticity. Novel solutions are presented for affine vorticity functions and a two-layer constant-vorticity scenario.

Ocean submesoscales, flows with characteristic size $10\,\text{m}{-}10\,\text{km}$, are transitional between the larger, rotationally constrained mesoscale and three-dimensional turbulence. In this paper, we present simulations of a submesoscale ocean filament. In our case, the filament is strongly sheared in both vertical and cross-filament directions, and is unstable. Instability indeed dominates the early behaviour with a fast extraction of kinetic energy from the vertically sheared thermal wind. However, the instability that emerges does not exhibit characteristics that match the perhaps expected symmetric or Kelvin–Helmholtz instabilities, and appears to be non-normal in nature. The prominence of the transient response depends on the initial noise, and for large initial noise amplitudes, saturates before symmetric instability normal modes are able to develop. The action of the instability is sufficiently rapid – with energy extraction from the mean flow emerging and peaking within the first inertial period ($\sim\! 18\ \text{h}$) – that the filament does not respond in a geostrophically balanced sense. Instead, at all initial noise levels, it later exhibits vertically sheared near-inertial oscillations with higher amplitude as the initial minimum Richardson number decreases. Horizontal gradients strengthen only briefly as the fronts restratify. These unstable filaments can be generated by strong mixing events at pre-existing stable structures; we also caution against inadvertently triggering this response in idealised studies that start in a very unstable state.

Bubbles entrained by breaking waves rise to the ocean surface, where they cluster before bursting and release droplets into the atmosphere. The ejected drops and dry aerosol particles, left behind after the liquid drop evaporates, affect the radiative balance of the atmosphere and can act as cloud condensation nuclei. The remaining uncertainties surrounding the sea spray emissions function motivate controlled laboratory experiments that directly measure and link collective bursting bubbles and the associated drops and sea salt aerosols. We perform experiments in artificial seawater for a wide range of bubble size distributions, measuring both bulk and surface bubble distributions (measured radii from $30\,\unicode{x03BC} \mathrm{m}$ to $5\,\mathrm{mm}$), together with the associated drop size distribution (salt aerosols and drops of measured radii from $50\,\mathrm{nm}$ to $500\,\unicode{x03BC} \mathrm{m}$) to quantify the link between emitted drops and bursting surface bubbles. We evaluate how well the individual bubble bursting scaling laws describe our data across all scales and demonstrate that the measured drop production by collective bubble bursting can be represented by a single framework integrating individual bursting scaling laws over the various bubble sizes present in our experiments. We show that film drop production by bubbles between $100\,\unicode{x03BC} \mathrm{m}$ and $1\,\mathrm{mm}$ describes the submicron drop production, while jet drop production by bubbles from $30\,\unicode{x03BC} \mathrm{m}$ to $2\,\mathrm{mm}$ describes the production of drops larger than $1\,\unicode{x03BC} \mathrm{m}$. Our work confirms that sea spray emission functions based on individual bursting processes are reasonably accurate as long as the surface bursting bubble size distribution is known.

The present work experimentally investigates the interaction of a buoyant (rigid) spherical particle with a single translating (water) vortex ring, focusing on the effects of particle-to-vortex core size ratio ($D_p/D_{c,o}$) on both the particle dynamics and ring dynamics ($D_p$ = particle diameter, $D_{c,o}$ = vortex core diameter). These interactions are studied for $D_p/D_{c,o}$ = 0.6–1.7, over ring Reynolds numbers ($Re={\varGamma }/{\nu }$; $\varGamma$ = ring circulation) of 6000–67 300. As the buoyant particle comes close to the ring, it gets captured into the low-pressure vortex core, and the interaction begins. The particle within the core undergoes radial oscillation, spins and translates along the ring’s azimuthal axis. As $D_p/D_{c,o}$ increases, the particle undergoes higher-amplitude radial oscillation and a relatively shorter azimuthal translation. The differences in the particle size and its motion within the ring lead to large differences in the ring’s dynamics. A larger particle is seen to lead to a higher ring disruption, substantially reducing the ring’s convection speed and azimuthal enstrophy, which are seen to scale as $(D_p/D_{c,o})^{2.3}Re^{-0.37}$ and $(D_p/D_{c,o})^{1.3}Re^{-0.25}$, respectively. The ring disruption is significant above $D_p/D_{c,o}\approx$ 1.0, beyond which the ring fragments, with up to 60 % drop in convection speed and 90 % drop in enstrophy, at low $Re$, as compared with the base ring. These results for the rigid particle size effects on the vortex ring dynamics are more dramatic than for a deforming bubble. Our results could help to better understand and model buoyant particle (and bubble) interactions with coherent structures in turbulence.

Volcanic fissure eruptions typically start with the opening of a linear fissure that erupts along its entire length, following which, activity localises to one or more isolated vents within a few hours or days. Localisation is important because it influences the spatiotemporal evolution of the hazard posed by the eruption. Previous work has proposed that localisation can arise through a thermoviscous fingering instability driven by the strongly temperature dependent viscosity of the rising magma. Here, we explore how thermoviscous localisation is influenced by the irregular geometry of natural volcanic fissures. We model the pressure-driven flow of a viscous fluid with temperature-dependent viscosity through a narrow fissure with either sinusoidal or randomised deviations from a uniform width. We identify steady states, determine their stability and quantify the degree of flow enhancement associated with localised flow. We find that, even for relatively modest variations of the fissure width (${\lt } 10$ %), the non-planar geometry supports strongly localised steady states, in which the wider parts of the fissure host faster, hotter flow, and the narrower parts of the fissure host slower, cooler flow. This geometrically driven localisation differs from the spontaneous thermoviscous fingering observed in planar geometries and can strongly impact the localisation process. We delineate the regions of parameter space under which geometrically driven localisation is significant, showing that it is a viable mechanism for the observed localisation under conditions typical of basaltic eruptions, and that it has the potential to dominate the effects of spontaneous thermoviscous fingering in these cases.

Phase change materials (PCMs) hold considerable promise for thermal energy storage applications. However, designing a PCM system to meet a specific performance presents a formidable challenge, given the intricate influence of multiple factors on the performance. To address this challenge, we hereby develop a theoretical framework that elucidates the melting process of PCMs. By integrating stability analysis with theoretical modelling, we derive a transition criterion to demarcate different melting regimes, and subsequently formulate the melting curve that uniquely characterises the performance of an exemplary PCM system. This theoretical melting curve captures the key trends observed in experimental and numerical data across a broad parameter space, establishing a convenient and quantitative relationship between design parameters and system performance. Furthermore, we demonstrate the versatility of the theoretical framework across diverse configurations. Overall, our findings deepen the understanding of thermo-hydrodynamics in melting PCMs, thereby facilitating the evaluation, design and enhancement of PCM systems.

We present the results of a theoretical investigation of orbital stability in pilot-wave hydrodynamics, wherein a droplet bounces and self-propels across the surface of a vertically vibrating liquid bath. A critical notion in pilot-wave hydrodynamics is that the bath plays the role of the system memory, recording the history of the droplet in its wave field. Quantised orbital motion may arise when the droplet is confined by either an axisymmetric potential or the Coriolis force induced by system rotation. We here elucidate the dependence of the stability of circular orbits on both the form of the confining force and the system memory. We first provide physical insight by distinguishing between potential- and wave-driven instabilities. We demonstrate that the former are a generic feature of classical orbital dynamics at constant speed, while the latter are peculiar to pilot-wave systems. The wave-driven instabilities are marked by radial perturbations that either grow monotonically or oscillate at an integer multiple of the orbital frequency, in which case they are said to be resonant. Conversely, for potential-driven wobbling, the instability frequency may be resonant or non-resonant according to the form of the applied potential. Asymptotic analysis rationalises the different stability characteristics for linear-spring and Coriolis forces, the two cases that have been explored experimentally. Our results are generalised to consider other potentials of interest in pilot-wave hydrodynamics, and elucidate the distinct roles of wave- and potential-driven instabilities. Our study highlights the limitations of prior heuristic arguments for predicting the onset of orbital instability.