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.

The thermocapillary flows generated by an inclined temperature gradient in and around a floating droplet are studied in the framework of the lubrication approximation. Numerical simulations of nonlinear flow regimes are fulfilled. It is shown that under the action of Marangoni stresses, a droplet typically moves as a whole. It is found that an inclined temperature gradient can lead to the excitation of periodic oscillations. With an increase of the inclination of the temperature gradient, temporally quasi-periodic oscillations have been obtained. In a definite region of parameters, an inclined temperature gradient can suppress oscillations, changing the droplet’s shape. The diagram of regimes in the plane of longitudinal and transverse Marangoni numbers has been constructed. Bistability has been found.

Spin coating is the process of generating a uniform coating film on a substrate by centrifugal forces during rotation. In the framework of lubrication theory, we investigate the axisymmetric film evolution and contact-line dynamics in spin coating on a partially wetting substrate. The contact-line singularity is regularized by imposing a Navier slip model. The interface morphology and the contact-line movement are obtained by numerical solution and asymptotic analysis of the lubrication equation. The results show that the evolution of the liquid film can be classified into two modes, depending on the rotational speed. At lower speeds, the film eventually reaches an equilibrium state, and we provide a theoretical description of how the equilibrium state can be approached through matched asymptotic expansions. At higher speeds, the film exhibits two or three distinct regions: a uniform thinning film in the central region, an annular ridge near the contact line, and a possible Landau–Levich–Derjaguin-type (LLD-type) film in between that has not been reported previously. In particular, the LLD-type film occurs only at speeds slightly higher than the critical value for the existence of the equilibrium state, and leads to the decoupling of the uniform film and the ridge. It is found that the evolution of the ridge can be well described by a two-dimensional quasi-steady analysis. As a result, the ridge volume approaches a constant and cannot be neglected to predict the variation of the contact-line radius. The long-time behaviours of the film thickness and the contact radius agree with derived asymptotic solutions.

Dean’s approximation for curved pipe flow, valid under loose coiling and high Reynolds numbers, is extended to study three-dimensional travelling waves. Two distinct types of solutions bifurcate from the Dean’s classic two-vortex solution. The first type arises through a supercritical bifurcation from inviscid linear instability, and the corresponding self-consistent asymptotic structure aligns with the vortex–wave interaction theory. The second type emerges from a subcritical bifurcation by curvature-induced instabilities and satisfies the boundary region equations. A connection to the zero-curvature limit was not found. However, by continuing from known self-sustained exact coherent structures in the straight pipe flow problem, another family of three-dimensional travelling waves can be shown to exist across all Dean numbers. The self-sustained solutions also possess the two high-Reynolds-number limits. While the vortex–wave interaction type of solutions can be computed at large Dean numbers, their branch remains unconnected to the Dean vortex solution branch.

Gravity-driven film flow in circular pipes with isolated topography was examined with fluorescence imaging for three flow rates, two angles of inclination, and four topography shapes. The time-averaged free surface response in the vicinity of the topography depended on flow rate, inclination angle and topography shape. For some flow conditions, the time-averaged free surface included a capillary ridge, and for a subset of those conditions, a series of capillary waves developed upstream with a spacing often approximated by half the capillary length. In contrast to film flow over planar topography, the capillary ridge often formed downstream of the topography, and for the lowest flow rate over rectangular step down topography, the free surface developed a steady overhang along the downstream face of the topography. Possible dynamic causes of the unique film flow behaviour in circular pipes are discussed. Transient free surface fluctuations were observed at half the magnitude reported in film flow over corrugated circular pipes, and local maxima in transient magnitude corresponded to axial locations of inflection points in the time-averaged free surface. Local maxima are related to low surface pressure regions that vary in location and amplitude. Rectangular step down topography generated an extra ridge of fluid that formed on top of the capillary ridge for flow conditions, resulting in a capillary ridge downstream of the step. The extra ridge varied in temporal duration and spatial extent, and finds no comparison in planar film flow. No evidence of periodic behaviour was detected in the transient film response.

Cilia perform various functions, including sensing, locomotion, generation of fluid flows and mass transport, serving to underpin a vast range of biological and ecological processes. However, analysis of the mass transport typically fails to resolve the near-field dynamics around individual cilia, and therefore overlooks the intricate role of power/recovery strokes of ciliary motion. Selvan et al. (2023, Phys. Rev. Fluids 8, 123103) observed that the flow field due to a point torque (i.e. a rotlet) accurately resolves both the near- and far-field characteristics of a single cilium’s flow in a semi-infinite domain. In this paper, we calculate the mass transport between a no-slip boundary and an adjacent fluid, as a model system for nutrient exchange with ciliated tissues. We develop a Langevin model in the presence of a point torque (i.e. a single cilium) to examine the nutrient flux from a localised surface source. This microscopic transport model is validated using a macroscopic continuum model, which directly solves the advection–diffusion equation. Our findings reveal that the flow induced by a point torque can enhance the particles’ transport, depending on their diffusivity and the magnitude of the point torque. Additionally, the average mass transport affected by a single cilium can be enhanced or diminished by the presence of an externally imposed linear shear flow, with a strong dependence on the alignment of the cilium. Taken together, this framework serves as a useful minimal model for examining the average nutrient exchange between ciliated tissues and fluid environments.

Two-dimensional gaseous detonations near critical propagation state were studied numerically in a channel with stoichiometric H$_2$/air and H$_2$/O$_2$ mixtures. Detonation waves exhibit a mode-locking effect (MLE) in a single-headed mode regime. Increasing the channel width alters the strength and propagation period of the single transverse wave. This leads to MLE failure and the occurrence of the single-dual-headed critical mode, featuring the emergence of a new transverse wave. For a stoichiometric H$_2$/air mixture, generation of the new transverse wave is due to interactions between the detonation front and the local explosion wave originating from interactions between the transverse wave and unreacted gas pocket downstream. Whereas, for a stoichiometric H$_2$/O$_2$ mixture, a transverse wave interacting with the wall produces Mach reflection bifurcation, causing MLE failure and generation of the new transverse wave. Our results show that all transverse waves manifest as strong transverse wave (STW) structures, with most belonging to the second kind, and an acoustic coupling exists between the typical second kind of STW structure and the acoustic wave in the induction zone behind the Chapman–Jouguet detonation front. A high-pressure region close to the STW structure plays a crucial role in exploring the transverse dynamics of this structure. Shock polars with rational assumptions are adopted to predict flow states in this region. The roles of pivotal factors in influencing the flow states and wave structure are clarified, and characteristic pressure values derived adequately represent the STW structure’s transverse dynamic behaviours. Lastly, the relationship between the kinematics and kinds of STW structures is unveiled.

The broad-band direct combustion noise is an important problem for industrial and domestic burners. The power spectral density (PSD) of this noise is related to the local spectral density of fluctuating heat release rate (HRR) ($\psi _{\dot {q}}$), which is challenging to measure but is readily available from large eddy simulations (LES) results. The behaviour of $\psi _{\dot {q}}$ for a wide range of thermochemical and turbulence conditions is investigated. Three burners are studied, namely a dual-swirl burner, a bluff-body burner and a jet in cross-flow burner, operating at atmospheric conditions with $\textrm {CH}_4$–air and $\textrm {H}_2$–air mixtures. In contrast to the classical $f^{-5/2}$ scaling, the far-field sound pressure level and volume-integrated HRR ($\psi _{\dot {Q}}$) spectra reveal a universal $f^{-5}$ scaling for high frequencies. This differing spectral decay rate for $\psi _{\dot {Q}}$ compared to the classical scaling is due to multi-regime combustion, related to either partial premixing or the local turbulence intensity. The dependence of $\psi _{\dot {q}}$ on the chosen spatial locations, flame configuration and its relation to velocity spectra are studied. A simple model for $\psi _{\dot {q}}$ involving the velocity spectra is found that compares well with LES results. The characteristic frequency involved in this model is related to the time scale of the coherent structures of the flow.

We examine the evaporation-induced coalescence of two droplets undergoing freezing by conducting numerical simulations employing the lubrication approximation. When two sessile drops undergo freezing in close vicinity over a substrate, they interact with each other through the gaseous phase and the simultaneous presence of evaporation/condensation. In an unsaturated environment, the evaporation flux over the two volatile sessile drops is asymmetric, with lower evaporation in the region between the two drops. This asymmetry in the evaporation flux generates an asymmetric curvature in each drop, which results in a capillary flow that drives the drops closer to each other, eventually leading to their coalescence. This capillary flow, driven by evaporation, competes with the upward movement of the freezing front, depending on the relative humidity in the surrounding environment. We found that higher relative humidity reduces the evaporative flux, delaying capillary flow and impeding coalescence by restricting contact line motion. For a constant relative humidity, the substrate temperature governs the coalescence phenomenon and the resulting condensation can accelerate this process. Interestingly, lower substrate temperatures are observed to facilitate faster propagation of the freezing front, which, in turn, restricts coalescence.

Manipulation of small-scale particles across streamlines is the elementary task of microfluidic devices. Many such devices operate at very low Reynolds numbers and deflect particles using arrays of obstacles, but a systematic quantification of relevant hydrodynamic effects has been lacking. Here, we explore an alternative approach, rigorously modelling the displacement of force-free spherical particles in vortical Stokes flows under hydrodynamic particle–wall interaction. Certain Moffatt-like eddy geometries with broken symmetry allow for systematic deflection of particles across streamlines, leading to particle accumulation at either Faxen field fixed points or limit cycles. Moreover, particles can be forced onto trajectories approaching channel walls exponentially closely, making possible quantitative predictions of particle capture (sticking) by short-range forces. This rich, particle-size-dependent behaviour suggests the versatile use of inertia-less flow in devices with a long particle residence time for concentration, sorting or filtering.

We perform direct numerical simulations of turbulent channel flows. Secondary motions are produced by applying a streamwise-homogeneous, spanwise-heterogeneous roughness pattern of spanwise period $\Lambda _s$ to the walls of the channel; their time evolution is observed. Notice that, owing to the geometry, the secondary motions are streamwise-invariant at any instant of time, so that no spatial development is seen. Once the secondary motions reach a statistically steady state, the roughness pattern is suddenly removed, so that the secondary motions decay. The time needed for the secondary motions to vanish is then measured; in doing so, we distinguish between the streamwise-momentum pathways and the cross-sectional circulatory motions that compose the secondary motions. Larger values of $\Lambda _s$ are generally associated with a longer time scale for the decay of the momentum pathways, although this might not hold true for $\Lambda _s/h\gt 4$ (where $h$ is the channel half-height). The value of such a time scale for the circulatory motions, instead, saturates for $\Lambda _s/h \geqslant 2$; this may be related to the observed spatial confinement of said circulatory motions. For specific values of $\Lambda _s$ ($2 \leqslant \Lambda _s/h \leqslant 4$), the volume-averaged energy associated with the momentum pathways undergoes an unexpected transient growth with respect to its value at the beginning of the decay. This might indicate that structures of such a specific size are able to self-sustain as postulated by Townsend (The Structure of Turbulent Shear Flow, 2nd edition, 1976, ch. 7.19); the evidence we gather in this respect is however inconclusive. Finally, the present data suggest that most of the energy of the momentum pathways is produced by the circulatory motions transporting the mean (spanwise-averaged) velocity.

Particle image velocimetry is used to study the control of swirl momentum, delivered through an orifice formed by a physically rotating tube of finite length, relevant to the evolution of vortex rings produced at a Reynolds number ${Re}\approx 1000$ based on the average discharge velocity, for swirl numbers ${S} \in [0, 1]$. Experiments without discharge, reinforced with complimentary numerical predictions, reveal the presence of an intriguing secondary flow pattern in the rotating tube, preventing attainment of a solid-body-like swirl distribution. Nevertheless, it is found that fully established rings produced in this way, following discharge once conditions in the tube have reached a steady state, exhibit similar characteristics to rings formed by an otherwise solid-body rotating initial condition as explored computationally by Ortega-Chavez et al. (2023, J. Fluid Mech. 967, A16). Namely, opposite-signed vorticity forms due to vortex tilting, which subsequently interacts with the ring, promoting vorticity cancellation and vortex ring breakdown. A key feature of the experimental work is that partially established vortex rings, produced before a steady-state rotating tube condition is reached, show unique characteristics. Their creation, a short time after the onset of tube rotation: (i) facilitates more efficient delivery of swirl momentum to the vortex core area; (ii) maintains a low level of swirl in the ring bubble’s central region which would otherwise promote the formation of opposite-signed vorticity and vortex breakdown.

We investigate the natural oscillations of sessile drops with a central trapped bubble on a plane using linear potential flow theory, considering both free and pinned contact lines. The system is governed by the contact angle $\alpha$ and the ratio $\tau$ of inner to outer contact line radii. For bubble-containing (BC) hemispherical drops with free contact lines (referred to as free BC semi-drops), the modes mirror half of those in concentric spherical BC drops due to plane symmetry. These modes are labelled ‘plus’ (with greater inner surface deformation) and ‘minus’ (with greater outer surface deformation). As $\tau \to 0$, minus modes converge to those of bubble-free drops. Results show that varying $\alpha$ from $90^\circ$ or pinning the contact line in free BC semi-drops alters the topology of spectral lines, turning original crossings of spectral lines between minus and plus modes into avoided crossings. This shift causes minus and plus modes to form spectral trends with avoided crossings, maintaining their original spectral shapes. In an avoided crossing, two coupled modes cannot be classified as plus or minus due to their comparable inner and outer surface deformations, resulting in mode beating when both are excited, as confirmed by our direct numerical simulations. This study on the impact of inner bubbles on the spectrum may help in predicting bubble size in opaque sessile drops.

Inertia–gravity waves are scattered by background flows as a result of Doppler shift by a non-uniform velocity. In the Wentzel–Kramers–Brillouin regime, the scattering process reduces to a diffusion in spectral space. Other inhomogeneities that the waves encounter, such as density variations, also cause scattering and spectral diffusion. We generalise the spectral diffusion equation to account for these inhomogeneities. We apply the result to a rotating shallow-water system, for which height inhomogeneities arise from velocity inhomogeneities through geostrophy, and to the Boussinesq system for which buoyancy inhomogeneities arise similarly. We compare the contributions that height and buoyancy variations make to the spectral diffusion with the contribution of the Doppler shift. In both systems, we find regimes where all contributions are significant. We support our findings with exact solutions of the diffusion equation and with ray tracing simulations in the shallow-water case.

We investigate fully developed turbulent flow in curved channels to explore the interaction between turbulence and curvature-driven coherent structures. By focusing on two cases of mild and strong curvature, we examine systematically the effects of the Reynolds number through a campaign of direct numerical simulations, spanning flow regimes from laminar up to the moderately high Reynolds number – based on bulk velocity and channel height – of $87\,000$. Our analysis highlights the influence of curvature on the friction coefficient, showing that flow transition is anticipated by concave curvature and delayed by convex curvature. In the case of mild curvature, a frictional drag reduction compared with plane channel flow is found in the transitional regime. Spectral analysis reveals that the near-wall turbulence regeneration cycle is maintained in mildly curved channels, while it is absent or severely inhibited on the convex wall of strongly curved channels. Streamwise large-scale structures resembling Dean vortices are found to be weakly dependent on the Reynolds number and strongly affected by curvature: increasing curvature shifts these vortices towards the outer wall and reduces their size and coherence, limiting their contribution to streamwise velocity fluctuations and momentum transport. In the case of strong curvature, spanwise large-scale structures are also detected. These structures are associated with large pressure fluctuations and the suppression of turbulent stresses near the convex wall, where a region with negative turbulence production is observed and characterised via quadrant analysis.

We introduce a wall model for large-eddy simulation (WMLES) applicable to rough surfaces with Gaussian and non-Gaussian distributions for both the transitionally and fully rough regimes. The model is applicable to arbitrary complex geometries where roughness elements are assumed to be underresolved, i.e. subgrid-scale roughness. The wall model is implemented using a multi-hidden-layer feedforward neural network, with the mean geometric properties of the roughness topology and near-wall flow quantities serving as input. The optimal set of non-dimensional input features is identified using information theory, selecting variables that maximize information about the output while minimizing redundancy among inputs. The model also incorporates a confidence score based on Gaussian process modelling, enabling the detection of potentially low model performance for untrained rough surfaces. The model is trained using a direct numerical simulation (DNS) roughness database comprising approximately 200 cases. The roughness geometries for the database are selected from a large repository through active learning. This approach ensures that the rough surfaces incorporated into the database are the most informative, achieving higher model performance with fewer DNS cases compared with passive learning techniques. The performance of the model is evaluated both apriori and aposteriori in WMLES of turbulent channel flows with rough walls. Over 550 channel flow cases are considered, including untrained roughness geometries, roughness Reynolds numbers and grid resolutions for both transitionally and fully rough regimes. Our rough-wall model offers higher accuracy than existing models, generally predicting wall shear stress within an accuracy range of 1%–15 %. The performance of the model is also assessed on a high-pressure turbine blade with two different rough surfaces. We show that the new wall model predicts the skin friction and the mean velocity deficit induced by the rough surface on the blade within 1%–10 % accuracy except the region with transition or shock waves. This work extends the building-block flow wall model (BFWM) introduced by Lozano-Durán & Bae (2023. J. Fluid Mech. 963, A35) for smooth walls, expanding the BFWM framework to account for rough-wall scenarios.

Large-eddy simulations are analysed to determine the influence of suspended canopies, such as those formed in macroalgal farms, on ocean mixed layer (OML) deepening and internal wave generation. In the absence of a canopy, we show that Langmuir turbulence, when compared with wind-driven shear turbulence, results in a deeper OML and more pronounced internal waves beneath the OML. Subsequently, we examine simulations with suspended canopies of varying densities located in the OML, in the presence of a background geostrophic current. Intensified turbulence occurs in the shear layer at the canopy’s bottom edge, arising from the interaction between the geostrophic current and canopy drag. Structures resembling Kelvin–Helmholtz (KH) instability emerge as the canopy shear layer interacts with the underlying stratification, radiating internal waves beneath the OML. Both intensified turbulence and lower-frequency motions associated with KH-type structures are critical factors in enhancing mixing. Consequently, the OML depth increases by up to a factor of two compared with cases without a canopy. Denser canopies and stronger geostrophic currents lead to more pronounced KH-type structures and internal waves, stronger turbulence and greater OML deepening. Additionally, vertical nutrient transport is enhanced as the OML deepens due to the presence of the canopy. Considering that the canopy density investigated in this study closely represents offshore macroalgal farms, these findings suggest a mechanism for passive nutrient entrainment conducive to sustainable farming. Overall, this study reveals the intricate interactions between the suspended canopy, turbulent mixing and stratification, underscoring their importance in reshaping OML characteristics.

A collection of secondary instability calculations in streaky boundary layers is presented. The data are retrieved from well-resolved numerical simulations of boundary layers forced by free-stream turbulence (FST), considering different geometries and FST conditions. The stability calculations are performed before streak breakdown, taking place at various $Rey_x$ the Reynolds number based on the streamwise coordinate. Despite the rich streak population of various sizes, it is found that breaking streaks have similar aspect ratios, independently of the streamwise position where they appear. This suggests that wider streaks will break down further downstream than thinner ones, making the appearance of secondary instabilities somewhat independent of the streak’s wavelength. Moreover, the large difference in the integral length scale among the simulations suggests that this aspect ratio is also independent of the FST scales. An explanation for this behaviour is provided by showing that these breaking streaks are in the range of perturbations that can experience maximum transient growth according to optimal disturbance theory. This could explain why, at a given streamwise position, there is a narrow spanwise wavelength range where streak breakdown is more likely to occur.