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.

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.

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.

The interaction between a turbulent flow and a porous boundary is analysed with focus on the sensitivity of the roughness function, $\Delta U^+$, to the upscaled coefficients characterizing the wall. The study is aimed at (i) demonstrating that imposing effective velocity boundary conditions at a virtual plane boundary, next to the physical one, can efficiently simplify the direct numerical simulations (DNS); and (ii) pursuing correlations to estimate $\Delta U^+$ a priori, once the upscaled coefficients are calculated. The homogenization approach employed incorporates near-interface advection via an Oseen-like linearization, and the macroscopic coefficients thus depend on both the microstructural details of the wall and a slip-velocity-based Reynolds number, $Re_{slip}$. A set of homogenization-simplified DNS is run to study the channel flow over transversely isotropic porous beds, testing values of the grains’ pitch within $0\lt \ell ^+\lt 40$. Reduction of the skin-friction drag is attainable exclusively over streamwise-aligned inclusions for $\ell ^+$ values up to $20{-}30$. The drag increase over spanwise-aligned inclusions (or streamwise-aligned ones at large $\ell ^+$) is accompanied by enhanced turbulence levels, including intensified sweep and ejection events. The root-mean-square of the transpiration velocity fluctuations at the virtual plane, $\tilde V_{rms}$, is the key control parameter of $\Delta U^+$; our analysis shows that, provided $\tilde V_{rms} \lesssim 0.25$, then $\tilde V_{rms}$ is strongly correlated to a single macroscopic quantity, $\Psi$, which comprises the Navier-slip and interface/intrinsic permeability coefficients. Fitting relationships for $\Delta U^+$ are proposed, and their applicability is confirmed against reference results for the turbulent flow over impermeable walls roughened with three-dimensional protrusions or different geometries of riblets.

The bi-stable dynamics of a one-degree-of-freedom disk pendulum swept by a flow and allowed to rotate in the cross-flow direction is investigated experimentally. For increasing flow velocity, a subcritical bifurcation is observed from a Pendulum state, characterised by an increasing time-averaged pendulum angle with large amplitude fluctuations, to a rotating state with a non-zero mean rotation velocity at a critical free stream velocity $U_{P2W}$. The rotating state, referred to as Windmill state, presents a strong hysteresis: once initiated, it is sustained down to velocities $U_{W2P}\lt U_{P2W}$ before bifurcating towards the Pendulum state. A thorough experimental characterisation of the dynamical features of each state is reported, with a particular focus on the influence of the static yaw angle of the disk $\beta _0$ and the free stream velocity. In the Pendulum state, the system behaves differently depending on whether $\beta _0$ lies below or above the stall angle of the disk, with more regular dynamics below. We demonstrate that the bifurcation between the Pendulum state and the Windmill state is triggered by aerodynamic fluctuations, while the bifurcation between the Windmill state and the Pendulum state is deterministic. A stochastic model faithfully reproduces the dynamical features of both states, as well as the characteristics of the bifurcations.

We present a linear analysis of a minimal model of moist convection under a variety of atmospheric conditions. The stationary solutions that we analyse include both fully saturated and partially unsaturated atmospheres in both unconditionally and conditionally unstable cases. We find that all of the solutions we consider are linearly unstable via exchange of stability when sufficiently driven. The critical Rayleigh numbers vary by over an order of magnitude between unconditionally unstable and conditionally unstable atmospheres. The unsaturated atmospheres are notable for the presence of linear gravity wave-like oscillations even in unstable conditions. We study their eigenfunction structure and find that the buoyancy and moisture perturbations are anticorrelated in $z$, such that regions of negative buoyancy have positive moisture content. We suggest that these features in unsaturated atmospheres may explain the phenomenon of gravity wave shedding by moist convective plumes.

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.

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 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.

Liquid metal flows are important for many industrial processes, including liquid metal batteries (LMBs), whose efficiency and lifetime can be affected by fluid mixing. We experimentally investigate flows driven by electrical currents in an LMB model. In our cylindrical apparatus, we observe a poloidal flow that descends near the centreline for strong currents, and a poloidal flow that rises near the centreline for weak currents. The first case is consistent with electrovortex flow, which is an interaction between current and its own magnetic field, whereas the second case is consistent with an interaction between current and the external field, which drives Ekman pumping. Notably, we also observe an intermediate case where the two behaviours appear to compete. Comparing results with Frick et al. (2022 J. Fluid Mech. 949, A20), we test prior estimates of the scaling of flow speed with current to predict the observed reversal. Based on these data, we propose two different ways to apply the Davidson et al. (1999 J. Fluid Mech. 245, 669–699) poloidal suppression theory that explain both experimental results simultaneously: either taking the wire radius into account to scale the Lorentz force, or taking viscous dissipation into account to scale the swirl velocity, following Herreman et al. (2021 J. Fluid Mech. 915, A17).

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.

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.

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.

Self-similar fractal tree models are numerically investigated to elucidate the drag coefficient, non-equilibrium dissipation behaviour and various turbulence statistics of fractal trees. For the simulation, a technique based on the lattice Boltzmann method with a cumulant collision term is used. Self-similar fractal tree models for aerodynamic computations are constructed using parametric L-system rules. Computations across a range of tree-height-based Reynolds numbers $Re_H$, from 2500 to 120 000, are performed using multiple tree models. As per the findings, the drag coefficients ($C_D$) of these models match closely with those of the previous literature at high Reynolds numbers ($Re_H \geqslant 60\,000$). A normalization process that collapses the turbulence intensity across various tree models is formulated. For a single tree model, a consistent centreline turbulence intensity trend is maintained in the wake region beyond a Reynolds number of 60 000. The global and local isotropy analysis of the turbulence generated by fractal trees indicates that, at high Reynolds numbers ($Re_H \geqslant 60\,000$), the distant wake can be considered nearly locally isotropic. The numerical results confirm the non-equilibrium dissipation behaviour demonstrated in previous studies involving space-filling fractal square grids. The non-dimensional dissipation rate $C_\epsilon$ does not remain constant; instead, it becomes approximately inversely proportional to the local Taylor-microscale-based Reynolds number, $C_\epsilon \propto 1/Re_\lambda$. We find significant one-point inhomogeneity, production and transverse transport of turbulent kinetic energy within the non-equilibrium dissipation near wake region.

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.

This article explores how a submerged elastic plate, clamped at one edge, interacts with water waves. Submerged elastic plates have been considered as potentially effective design elements in the development of wave energy harvesters but their behaviour in a wave field remains largely unexplored, especially experimentally. Positioned at a fixed depth in a wave tank, the flexible plate demonstrates significant wave reflection capabilities, a characteristic absent in rigid plates of identical dimensions. The experiments thus reveal that plate motion is crucial for wave reflection. Sufficiently steep waves are shown to induce a change in the mean position of the plate, with the trailing edge reaching the free surface in some cases. This configuration change is found to be particularly efficient to break water waves. These findings contribute to understanding the potential of elastic plates for wave energy harvesting and wave attenuation scenarios.

Wave propagation in channels with area changes is a topic of significant practical interest that involves a rich set of coupled physics. While the acoustic wave problem has been studied extensively, the shock propagation problem has received less attention. In addition to its practical significance, this problem also introduces deep fundamental issues associated with how energy in propagating large-amplitude disturbances is redistributed upon interaction with inhomogeneities. This paper presents a study of shock scattering and entropy and vorticity coupling for shock wave propagation through discrete area changes. It compares results from computational fluid dynamics to those of one-dimensional quasi-steady calculations. The solution space is naturally divided into five ‘regimes’ based upon the incident shock strength and area ratio. This paper also presents perturbation methods to quantify the dimensionless scaling of physical effects associated with wave reflection/transmission and energy transfer to other disturbances. Finally, it presents an analysis of the ‘energetics’ of the interaction, quantifying how energy that initially resides in dilatational disturbances and propagates at the shock speed is redistributed into finite-amplitude reflected and transmitted waves as well as convecting vortical and entropy disturbances.

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.

The influence of irregular three-dimensional rough surfaces on the displacement of the logarithmic velocity profile relative to that of a smooth wall in turbulent flow, known as the roughness function, is studied using direct numerical simulations. Five different surface power spectral density (PSD) shapes were considered, and for each, several rough Gaussian surfaces were generated by varying the root mean square ($k_{rms}$) of the surface heights. It is shown that the roughness function ($\Delta U^{+}$) depends on both the PSD and $k_{rms}$. For a given $k_{rms}$, $\Delta U^{+}$ increases as the wavenumbers of the PSD expand to large values, but at a rate that decreases with the magnitude of the wavenumbers. Although $\Delta U^{+}$ generally does not scale with either $k_{rms}$ or the effective slope $ES$ when these variables are considered singularly, for PSDs with low wavenumbers, $\Delta U^{+}$ tends to scale with $ES$, whereas as wavenumbers increase, $\Delta U^{+}$ tends to scale with $k_{rms}$. An equivalent Nikuradse sand roughness of about eight times $k_{rms}$ is found, which is similar to that observed in previous studies for a regular three-dimensional roughness. Finally, it is shown that $k_{rms}$ and the effective slope are sufficient to describe the roughness function in the transitional rough regime.

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.