In this study, we conducted interface-capturing high-resolution simulations of a bubbly upflow in a vertical channel to investigate the bubble distribution and its interaction with surrounding turbulence, focusing on the effects of the density ratio. A bulk Reynolds number $Re_b=2300$ was used for all simulations. The influence of density ratio on vortex structures and turbulence statistics differed between the near-wall and core regions of the channel. Adding 5.43 $\%$ gas caused an increase in wall friction. By applying a generalised FIK identity to analyse wall friction, it was determined that the drag rise in the bubbly channel was mostly due to the near-wall region. Visualisation of the bubble and vortex structures showed that small bubbles near the wall induced larger magnitude of Reynolds shear stress and increased wall friction. Bubble behaviour near the wall region was similar for density ratios above 30, leading to wall friction saturation. In the core region, large deformable bubbles created wake vortices due to slip velocity between liquid and gas phases. Wake vortices help large bubbles absorb smaller bubbles and maintain their sizes. As the density ratio increased, the slip velocity increased owing to greater difference in the gravitational acceleration between liquid and gas phases, resulting in corresponding increase in wake intensity and velocity fluctuations. However, quadrant analysis showed that Q1 and Q3 events increased together with Q2 and Q4 events in the core region, cancelling out any net effect of wake vortices on Reynolds shear stress or wall friction.

Highly resolved simulations reveal the fundamental influence of a carrier fluid’s flow dynamics on triboelectric powder charging. We found that particles transported through a square-shaped duct charge faster than in a channel flow caused by secondary flows that led to more severe particle–wall collisions. Specifically, particles with a Stokes number of 4.69 achieve 85 % of their equilibrium charge approximately 1.5 times faster in duct flow than in channel flow. Also, charge distribution is more uniform in a duct cross-section compared with a channel cross-section. In channel flow, particles are trapped near the walls and collide frequently due to limited movement in the wall-normal direction, causing localized charge buildup. In contrast, duct flow promotes better mixing through secondary flows, reducing repeating collisions and providing uniform charge distribution across the cross-section. Upon charging, electrostatic forces significantly reshape particle behaviour and distribution. Once the powder achieves half of its equilibrium charge, particles increasingly accumulate at the wall, leading to a reduced concentration in the central region. These changes in particle distribution have a noticeable impact on the surrounding fluid phase and alter the overall flow dynamics. These findings open the possibility for a new measure to control powder charging by imposing a specific pattern.

Real-time wave forecasting (RTWF) consists in predicting ocean wave motion or forces, from seconds to minutes in advance, using real-time measurements. For the successful development of RTWF, understanding wave predictability is essential. Usually, a deterministic ‘predictable zone’ (DPZ) is geometrically constructed from the wave group velocities and directions present in the spectrum. DPZs have little experimental evidence, and suffer ambiguities regarding the choice of cutoff frequencies and directions – since actual ocean waves are not band-limited. The present study addresses those shortcomings, by defining probabilistic predictable zones (PPZs) with respect to chosen uncertainty thresholds, using a rigorous statistical framework restricted to near-Gaussian sea states (precisely those where RTWF would be employed). PPZs are examined in idealised spectra and in a stereo dataset of a real wave field. It is shown that the PPZ geometry is quantitatively related to the sea state characteristics, through three physical parameters: two limiting group velocities (similar to the deterministic theory), and a directional spreading effect, which also limits the PPZ extent. While the lower group velocity depends on the chosen uncertainty threshold, the upper group velocity is better approximated by that of the spectrum peak frequency, which is a novel finding. The empirical data support the validity of the present PPZ theory. In contrast, both theoretical and empirical results contradict the fan-shaped predictable zones, constructed in the three-dimensional deterministic theory, thus highlighting the importance of considering stochasticity to understand the predictability of actual ocean waves.

We report laboratory experiments of long-crested irregular water surface waves propagating over a shoal, with attention to the region over the down-slope behind the shoal. We measure the surface elevation field, the horizontal velocity field in the water, and the resulting forces on a horizontal submerged cylinder placed over the down-slope of the shoal. In addition, we calculate the horizontal acceleration field. From this, we find that the presence of the shoal can modify the wave field such that the resulting forces on the submerged cylinder can be enhanced with thicker extreme tails and increased values of skewness and kurtosis depending on the location of the cylinder. The spatial dependence of the statistics of forces is different from the spatial dependence of the statistics of horizontal velocity, horizontal acceleration and surface elevation.

A pressure-gradient-induced laminar separation bubble (LSB) was examined using wind-tunnel experiments, direct numerical simulations (DNS) and linear local/global stability analysis. The LSB was experimentally generated on a flat plate using the favourable-to-adverse pressure gradient imposed by an inverted modified NACA $64_3-618$ airfoil. Direct numerical simulation was performed using boundary conditions extracted from a steady precursor simulation of the entire flow field. Despite good agreement in the upstream boundary layer with the experiment, DNS exhibited an approximately 25 % longer mean separation bubble, attributed to an earlier onset of transition due to the free-stream turbulence (FST) in the experiment. Introducing a very low level of isotropic FST in the DNS, similar to that measured in the experiment, caused earlier transition, decreased the mean bubble length and led to a remarkably good agreement between the DNS and experiments. Differences were observed for the dominant frequencies in the experiment and DNS, but both were within the band of most amplified frequencies predicted by LST. Proper orthogonal decomposition confirmed that dominant coherent structures from DNS and experiments are related to the inviscid Kelvin–Helmholtz instability and have similar characteristics despite slight differences in frequency. Local and global stability and dynamic mode decomposition analysis corroborated the convective nature of the dominant two-dimensional (2-D) instability and showed that the LSB is globally unstable to a range of 3-D wavenumbers, in agreement with 3-D structures observed in experiments. Results confirmed the strong impact of very low FST levels on the LSB and indicate a close agreement of the time-averaged and instability characteristics between the experiments and DNS.

The incompressible Navier–Stokes equations in spherical coordinates are solved using a pseudo-spectral method to simulate the problem of spherical Couette flow. The flow is investigated for a narrow-gap ratio with only the inner sphere rotating. We find that the flow is sensitive to the initial conditions and have used various initial conditions to obtain different branches of the bifurcation curve of the flow. We have identified three different branches dominated respectively by axisymmetric flow, travelling wave instability and equatorial instability. The axisymmetric branch shows unsteadiness at large Reynolds numbers. The travelling wave instability branch shows spiral instability and is prominent the near poles. The travelling wave instability branch further exhibits a reversal in the propagation direction of the spiral instability as the Reynolds number is increased. This branch also exhibits a multi-mode equatorial instability at larger Reynolds numbers. The equatorial instability branch exhibits twin jet streams on either side of the equator, which become unstable at larger Reynolds numbers. The flow topology on the three branches is also investigated in their phase space and found to exhibit chaotic behaviour at large Reynolds numbers on the travelling wave instability branch.

Combustion instability analysis in annular systems often relies on reduced-order models that represent the complexity of combustion dynamics in a framework in which the flame is represented by a ‘flame describing function’ (FDF), portraying its heat release rate response to acoustic disturbances. However, in most cases, FDFs are only available for a limited range of disturbance amplitudes, complicating the description of the saturation process at high oscillation levels leading to the establishment of a limit cycle. This article shows that this difficulty may be overcome using a novel experimental scheme, relying on injector staging and in which the oscillation amplitude at limit cycle can be controlled, enabling us to measure FDFs from simultaneous pressure and heat release rate recordings. These data are then exploited to replace the standard modelling, in which the heat release rate is expressed as a third-order polynomial of pressure fluctuations, by a function of the modulation amplitude, allowing an easier adaptation to experimental data. The FDF is then used in a dynamical framework to analyse a set of staging configurations in an annular combustor, where two families of injectors are mixed and form different patterns. The limit-cycle amplitudes and the coupling modes observed experimentally are suitably retrieved. Finally, an expression for the growth rate is derived from the slow-flow variable equations defining the modal amplitudes and phase functions, which is shown to exactly agree with that obtained previously by using acoustic energy principles, providing a theoretical link between growth rates and limit-cycle amplitudes.

A generalized reciprocal theorem is used to relate the force and torque induced on a particle in an inertia-less fluid with small variation in viscosity to integrals involving Stokes flow fields and the spatial dependence of viscosity. These resistivity expressions are analytically evaluated using spheroidal harmonics and then used to obtain the mobility of the spheroid during sedimentation, and in linear flows, of a fluid with linear viscosity stratification. The coupling between the rotational and translational motion induced by stratification rotates the spheroid’s centerline, creating a variety of rotational and translational dynamics dependent upon the particle’s aspect ratio, $\kappa$, and the component of the stratification unit vector in the gravity direction, $d_g$. Spheroids with $0.55\lessapprox \kappa \lessapprox 2.0$ exhibit the largest variety of settling behaviors. Interestingly, this range covers most microplastics and typical microorganisms. One of the modes include a stable orientation dependent only on $\kappa$ and $d_g$, but independent of initial orientation, thus allowing for the potential control of settling angles and sedimentation rates. In a simple shear flow, cross-streamline migration occurs due to the stratification-induced force generated on the particle. Similarly, a particle no longer stays at the stagnation point of a uniaxial extensional flow. While fully analytical results are obtained for spheroids, numerical simulations provide a source of validation. These simulations also provide additional insights into the stratification-induced force- and torque-producing mechanisms through the stratification-induced stress, which is not accessed in the reciprocal theorem-based analytical calculations.

We investigate the stability of a compressible boundary layer over an impedance wall for both constant impedances and a frequency-dependent porous wall model. For an exponential mean flow profile, the solution of the Pridmore-Brown equation, i.e. the linearised Euler equations for compressible shear flows, is expressed exactly in confluent Heun functions and, with the boundary condition of acoustic wall impedance, reduced to a single algebraic eigenvalue equation. This, in turn, is solved asymptotically and numerically and provides the complete inviscid eigenvalue spectrum without spurious modes. The key finding is that impedance walls not only have a desirable stabilising effect on inviscid disturbances, but also induce new instabilities. The type of the destabilised mode and therefore also the direction of propagation of the modes with maximum growth rate as well as the destabilised wavenumbers depend significantly on the porous wall properties, in particular on the porous wall layer thickness. For small porous layer thicknesses, the impedance-induced instability is observed as a second mode instability, where we find above a critical porosity growth rates exceeding those present in the rigid-wall case.

Turbulence beneath a free surface leaves characteristic long-lived signatures on the surface, such as upwelling ‘boils’, near-circular ‘dimples’ and elongated ‘scars’, easily identifiable by eye, e.g. in riverine flows. In this paper, we analyse data from direct numerical simulations to explore the connection between these surface signatures and the underlying vortical structures. We investigate dimples, known to be imprints of surface-attached vortices, and scars, which have yet to be extensively studied, by analysing the conditional probabilities that a point beneath a signature is within a vortex core as well as the inclination angles of sub-signature vorticity. The analysis shows that the probability of vortex presence beneath a dimple decreases from the surface down through the viscous and blockage layers. This vertical variation in probability is approximately a Gaussian function of depth and depends on the dimple’s size and the bulk turbulence properties. Conversely, the probability of finding a vortex beneath a scar increases sharply from the surface to a peak at the edge of the viscous layer, regardless of scar size. The probability distributions of the angle between the vorticity vector and the vertical axis also show a clear pattern about vortex orientation: a strong preference for vertical alignment below dimples and an equally strong preference for horizontal alignment below scars. Our findings corroborate previous studies that tie dimples to surface-attached vertical vortices. Moreover, they suggest that scars can be defined as imprints of horizontal vortices that are located approximately a quarter of the Taylor microscale beneath the free surface.

The encapsulation of active particles, such as bacteria or active colloids, inside a droplet gives rise to a non-trivial shape dynamics and droplet displacement. To understand this behaviour, we derive an asymptotic solution for the fluid flow about a deformable droplet containing an active particle, modelled as a Stokes-flow singularity, in the case of small shape distortions. We develop a general solution for any Stokes singularity and apply it to compute the flows and resulting droplet velocity due to common singularity representations of active particles, such as Stokeslets, rotlets and stresslets. The results show that offsetting of the active particle from the centre of the drop breaks symmetry and excites a large number of generally non-axisymmetric shape modes as well as particle and droplet motion. In the case of a swimming stresslet singularity, a run-and-tumble locomotion results in superdiffusive droplet displacement. The effect of interfacial properties is also investigated. Surfactants adsorbed at the droplet interface counteract the internal flow and arrest the droplet motion for all Stokes singularities except the Stokeslet. Our results highlight strategies to steer the flows of active particles and create autonomously navigating containers.

The evolution of two-phase structures, turbulence/dust concentration structures, during an entire sandstorm process, including non-stationary flow, has been originally investigated in this study. Dust concentration structures are observed at different sandstorm stages, which are similar to the turbulence structures. These two-phase structures adhere to self-similarity in the steady stage but fail in the non-stationary stage. However, dust particle exhibits a better capability to follow eddies in flow, but the evolution of dust structures is not analogous to that of turbulence structures, exhibiting distinct trends. Dust particles, initiated from the ground, gradually form cluster structures in the rising stage. Their morphology exhibits a ridge-like evolutionary trend, reaching a peak in the steady stage. In contrast, turbulence structures are most persistent and oblique in the early stage but sequentially diminish in the subsequent steady and declining stages. The significant changes in shear due to sharply varying wind velocity and thermal stability are primarily responsible for these evolution differences.

We revisit viscoelastic Kolmogorov flow to show that the elastic linear instability of an Oldroyd-B fluid at vanishing Reynolds numbers ($Re$) found by Boffetta et al. (J. Fluid Mech., vol. 523, 2005, pp. 161–170) is the same ‘centre-mode’ instability found at much higher $Re$ by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502) in a pipe and by Khalid et al. (J. Fluid Mech., vol. 915, 2021, A43) in a channel. In contrast to these wall-bounded flows, the centre-mode instability exists even when the solvent viscosity vanishes (e.g. it exists in the upper-convective Maxwell limit with $Re=0$). Floquet analysis reveals that the preferred centre-mode instability almost always has a wavelength twice that of the forcing. All elastic instabilities give rise to familiar ‘arrowheads’ (Page et al., Phys. Rev. Lett., vol. 125, 2020, 154501) which in sufficiently large domains and at sufficient Weissenberg number ($W$) interact chaotically in two dimensions to give elastic turbulence via a bursting scenario. Finally, it is found that the $k^{-4}$ scaling of the kinetic energy spectrum seen in this two-dimensional elastic turbulence is already contained within the component arrowhead structures.

The present work proposes a general analysis of those models for gravity wave propagation that partially or totally rely on an average procedure over the water depth. The aim is the identification of the intrinsic physical quantities that characterize the wave dynamics, going beyond the usual definition of depth-averaged velocity. In particular, the proposed approach is based on the decomposition of the depth-averaged fields in their gradient- and divergence-free components. This naturally leads to the definition of a generalized velocity field that includes part of the dispersive contributions of the wave dynamics, and to the detection of the intrinsic boundary conditions along the free surface and the seabed. The analysis also proves the existence of generalized velocity potentials that under particular circumstances can include rotational contributions.

We study the dynamics of fracture deflation following hydraulic fracturing of an infinite elastic solid, with fluid removal from a narrow conduit at the centre. This process involves coupled lubricating flow and elastic deformation, now subject to appropriate descriptions of fluid removal through the conduit towards the ambient, driven by elastic stresses and extraction/suction. When the influence of material toughness is negligible, the dynamics is found to be governed by two dimensionless parameters that describe the relative influence of elasticity-driven backflow ($\Pi _c$) and ambient-pressure-driven backflow ($\Pi _e$), respectively. We also found that the fracture’s thickness eventually approaches zero at the centre, while the fracture evolves into a self-similar shape of the dipole type that conserves the dipole moment $M$. The fracture’s front continues to elongate according to $x_f \propto t^{1/9}$, while the total fluid volume within the fracture decreases according to $V \propto t^{-1/9}$. The model and solutions might find use in practical problems to estimate the rate of backflow and effective permeability of a fractured reservoir once pressure is released.

The aerodynamic deformation and breakup of wall-attached droplets in axisymmetric stagnation flow are investigated experimentally. A vertical shock tube is used to generate the shock wave accompanying the post-wave airflow, and the axisymmetric stagnation flow is formed through the impingement of an air stream on a solid wall. For the wall-attached droplets with initially hemispherical profile, four typical droplet deformation and breakup modes can be identified with the continuous increase of the droplet local Weber number, which are the vibrating mode, the compressing mode, the sheet thinning mode and the shear-induced entrainment mode. Quantitative analyses of droplet evolution dynamics are also conducted for the compressing mode and the sheet thinning mode, and the significant differences of air flow separation at the droplet lateral surface between these two modes are revealed. The potential flow model and the energy conservation model are further developed to predict the entire droplet deformation processes. The vibrating frequency and amplitude of droplets under the vibrating mode are predicted by a spring-mass model, and the surface perturbation wavelengths of droplets under the shear-induced entrainment mode are estimated based on the dispersion relation of Kelvin–Helmholtz instability. This work is proposed to give potential guidance for regulating the aerodynamic fragmentation of wall-attached droplets in practical engineering applications.

Supersonic impinging tones have been attracting significant interest because high-intensity discrete-frequency tones pose substantial risks to structural safety in applications such as rocket launch and recovery, and space vehicle attitude adjustment. However, various issues remain to be addressed regarding the jet oscillation and tone generation mechanism. In this study, a numerical simulation of the supersonic impinging jet with a nozzle pressure ratio of 4.03 and an impingement distance of 2.08 times the nozzle exit diameter is conducted. The results show good consistency with the reference data by other researchers. A phase-locked averaging analysis of 2960 flow field snapshots is employed to investigate jet structure oscillation dynamics and the tone generation mechanism. The phase-locked averaged images reveal that the pressure variation induced by Kelvin–Helmholtz vortices as they pass through the reflected shock results in the periodic motions of the reflected shock and Mach disk. The periodically oscillating Mach disk generates high-pressure fluid masses driving recirculation bubbles through a cyclic ‘compression–generation–merging’ oscillation. The streamline oscillation and sound-ray analyses reveal there are two distinct tone source regions: the impinging zone and the wall jet region. Consequently, it is proposed that vortex collapse in conjunction with wall jet oscillations coexist to generate the tone. According to the directivity, the tone emitted from the wall jet source region is believed to contribute to the feedback loop. These findings collectively contribute to an improved understanding of the jet plume oscillation and tone generation mechanisms of the supersonic impinging jet.

The dynamics of a stratified fluid in which the rotation vector is slanted at an angle with respect to the local vertical (determined by gravity) is considered for the case where the aspect ratio of the characteristic vertical scale of the motion D to the horizontal scale L is not small. In cases where the Rossby number of the flow is small the natural coordinate system is non-orthogonal and modifications to the dynamics are significant. Two regimes are examined in this paper. The first is the case in which the horizontal length scale of the motion, L, is sub-planetary where the quasi-geostrophic approximation is valid. The second is the case where the horizontal scale is commensurate with the planetary radius and so the dynamics must be formulated in spherical coordinates with imposing a full variation on the relevant components of rotation. In the quasi-geostrophic case the rotation axis replaces the direction of gravity as the axis along which the geostrophic flow varies in response to horizontal density gradients. The quasi-geostrophic potential vorticity equation is most naturally written in a non-orthogonal coordinate system with fundamental alterations in the dynamics. Examples such as the reformulation of the classical Eady problem are presented to illustrate the changes in the nature of the dynamics. For the second case where the horizontal scale is of the order of R, the planetary radius, more fundamental changes occur leading to more fundamental and difficult changes in the dynamical model.

We study nonlinear resonant triad interactions among flexural-gravity waves generated by a steadily moving load on a floating ice sheet. Of the many possible triad interactions involving at least one load-produced wave, we focus on the double-frequency case where the wavenumber of the leading wave is double that of the trailing wave. This case stands out because resonant interactions can occur with or without the presence of an ambient wave. Using multiple-scale perturbation analysis, we obtain amplitude evolution equations governing the leading-order, steady-state response. We complement the theoretical predictions with direct numerical simulations of the initial–boundary value problem using a high-order spectral method accurate to arbitrary order. Our results show that the double-frequency interaction can cause the trailing wave amplitude to decay with distance from the load, with its energy transferred to its second harmonic which radiates forwards to coherently interfere with the leading wave. Depending on the length and orientation of the load, the resonant interaction can in some cases cause the wave drag to become vanishingly small, or in other cases nearly double the maximum bending strain compared to the linear prediction. We also consider the effect of a small ambient wave that can initiate a resonant interaction in the leading wave field in addition to the trailing wave field interaction. This can result in a steady, localised wave packet containing two mutually trapped wave components, leading to vanishing wave drag. This work has potential implications for defining safe operating profiles for vehicles travelling on floating ice.