Edge waves, the infragravity waves trapped by near-shore topography, are important in morphodynamics and flooding on mildly sloping beaches. Edge waves are usually generated by swell via triad interactions. Here, we examine the possibility that edge waves might be also generated directly by wind. By processing data from the SandyDuck’97 near-shore experiment, we show that pronounced directional asymmetry of edge waves does occur in nature, apparently unrelated to the direction of swells and along-shore currents. These observations exhibit edge waves propagating in the downwind direction under moderate wind against the along-shore currents, while swell is incident nearly normally to the shoreline, which strongly suggests generation of edge waves by wind. We examine theoretically possible mechanisms of edge-wave excitation by wind. We show that the ‘maser’ mechanism suggested by Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 311, issue 1506, 1969b, pp. 371–389) in the context of excitation of free water waves is effective under favourable conditions: nonlinearly interacting random short wind-forced waves create a viscous shear stress on the water surface with the variation of stress being phase linked to edge waves, which allows self-excitation of a coherent edge wave. The model we put forward is based upon the kinetic equation for short wind waves propagating on the inhomogeneous current due to an edge wave. The model needs a dedicated experiment for validation. Analysis of plausible alternative mechanisms of generation via Miles’ critical layer and via the viscous shear stresses induced by the edge wave in the air revealed no instability in the consideration confined to the main mode and constant slope bathymetry.

The distinct turbulence dynamics and transport modulated by a common seagrass species were investigated experimentally using a flexible surrogate canopy in a refractive-index-matching environment that enabled full optical access. The surrogate seagrass replicated the dynamic behaviour and morphological properties of its natural counterpart. The flows studied were subcritical with Froude numbers $Fr<0.26$ and concerned five Reynolds numbers $Re\in [3.4\times 10^{4}, 1.1\times 10^{5}]$ and Cauchy numbers $Ca\in [120, 1200]$. Complementary rigid canopy experiments were also included to aid comparative insight. The flow was quantified in wall-normal planes in a developed region using high-frame-rate particle image velocimetry. Results show that the deflection and coordinated waving motion of the blades redistributed the Reynolds stresses above and below the canopy top. Critically, in-canopy turbulence associated with the seagrass lacked periodic stem wake vortex shedding present in the rigid canopy, yet the flexible canopy induced vortex shedding from the blade tips. Inspection of spatial and temporal characteristics of coherent flow structures using spectral proper orthogonal decomposition reveals that Kelvin–Helmholtz-type vortices are the dominant flow structures associated with the waving motion of the seagrass and that modulated the local flow exchange in both rigid and flexible canopies. A barrier-like effect produced by the blade deflections blocked large-scale turbulence transport, thereby reducing vortex penetration into the canopy. In addition, we uncovered a transition from sweep-dominated to ejection-dominated behaviour in the surrogate seagrass. We hypothesise that the vortices created during the upward blade motion period play a major role in the sweep-to-ejection-dominated transition. Conditionally averaged quadrant analysis on the downward and upward blade motion supports this contention.

We investigate experimentally the dynamics of non-axisymmetric fibres in channel flow turbulence, focusing specifically on the importance of the fibre size relative to the flow scales. To this aim, we maintain the same physical size of the fibres and we increase the shear Reynolds number. Experiments are performed in the TU Wien Turbulent Water Channel for three values of shear Reynolds number, namely 180, 360 and 720. Fibres are slender – length to diameter ratio of 120 – rigid, curved and neutrally buoyant particles and their shape ranges from low curvature – almost straight fibres – to moderate curvature. In all cases, fibre size remains small compared with the channel height (${\leqslant }1.5\,\%$). Three-dimensional and time-resolved recordings of the laser-illuminated measurement region are obtained from four high-speed cameras and used to infer fibre dynamics. With the aid of multiplicative algebraic reconstruction techniques, fibre position, orientation, velocity and rotation rates are determined. Our measurements span over the half-channel height, from wall to centre, and allow a complete characterisation of the fibre dynamics in all regions of the flow. Specifically, we measure fibre preferential distribution and orientation. We observe that the fibre dynamics is always influenced by their curvature. Through a comparison between measurements of the near-wall dynamics of the fibres and the near-wall dynamics of the flow, we identify a causal relationship between fibre velocity and orientation, and the near-wall turbulence dynamics. Finally, we have been able to provide original measurements of the tumbling rate of the fibres, for which we report the influence of fibre curvature. We underline that our measurements confirm previous findings obtained in numerical and experimental works.

We describe wall-resolved and wall-modelled large-eddy simulation (LES) of plane Couette (PC) flow. Subgrid-scale (SGS) motion is represented using the stretched-spiral vortex SGS model and the virtual wall model is employed for wall-modelled LES. Cases studied include direct numerical simulation (DNS) at friction Reynolds numbers $Re_\tau = 220$, wall-resolved LES at $Re_\tau \sim 500\text {--}3600$ and wall-modelled LES at $Re_\tau \sim 3600\text {--}2.8\times 10^{5}$. All LES performed show the presence of approximately spanwise periodic sets of streamwise rolls. Averaged (including spanwise) wall-normal profiles of the mean streamwise velocity show a consistent log region across all Reynolds numbers. Two distinct measures of turbulent intensity are explored, one of which recognizes the roll structure and one that does not. The spanwise variation of turbulence flow metrics is investigated. Mean streamwise velocity profiles show substantial spanwise variation but collapse well when normalized by local skin-friction velocities. Similar collapse is found for streamwise turbulent intensities. For all present LES, the mean skin-friction variation with the plate Reynolds number is found to match a simple analytical form (Pirozzoli et al., J. Fluid Mech., vol. 758, 2014, pp. 327–343) while the scaled centre-plane, mean-velocity gradient exhibits an inverse ProductLog dependence. Both the mean-flow roll energy and circulation, scaled with outer variables, decrease monotonically for $Re_\tau \gtrsim 500$. At lower $Re_\tau$, the mean streamwise zero-velocity line follows a wavy form in the spanwise direction, while at our larger $Re_\tau$, a mushroom shape emerges which could potentially enhance local momentum transport in the spanwise direction and be responsible for the weakening of the spanwise rolls.

Chemically active droplets often show intriguing self-propulsion behaviour in a surfactant solution. The drop motion is controlled by the nonlinear coupling among chemical transport in the bulk fluid, consumption of surfactant at the drop surface, and the fluid flow driven by the self-generated Marangoni stress. To quantify the underlying hydrodynamics, this work investigates the swimming motion of a two-dimensional drop that is determined by two dimensionless parameters, the Péclet number ($Pe$) and Damköhler number ($Da$). The weakly nonlinear analysis shows that near the instability threshold, the drop undergoes a supercritical bifurcation with velocity $U\sim \sqrt {Pe-Pe_c}$, where $Pe_c$ is the critical Péclet number for the onset of dipole mode. In the highly nonlinear regime, the drop transits from steady translation of pusher swimming to unsteady motion of mixed pusher–puller swimming along zigzaging trajectories of quadrangle and/or triangle waves. Mode decomposition shows that the zigzag motion is directly related to the interaction between the secondary wake of low surfactant concentration and the primary wake.

This study examines the effects of an evolving attractant field, due to the phenomenon of autochemotaxis, on the stability of a dilute suspension of active swimmers in a channel. Motile swimmers typically exhibit an orientation bias that is dependent upon the local gradient in the attractant field. By modelling the autochemotactic behaviour of the swimmers through a non-dimensional number $Da$ that characterizes the rate of attractant secretion, we are able to show that the active stress-driven instability predicted by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, p. 038101) for a suspension of pushers is stabilized with increasing $Da$. We also show that the autochemotactic behaviour results in an active stress-driven flow instability for a suspension of pullers. Furthermore, we show analytically and numerically that in the absence of convective transport of the attractant, pushers and pullers undergo a simultaneous switch of stabilities at a critical $Da$.

A direct numerical simulation database of a weakly compressible turbulent channel flow with bulk Mach number 1.56 is studied in detail, including the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor, as well as the mechanism of the pressure-Hessian tensor contributing to the evolution of invariants of the velocity gradient tensor. The results show that the geometrical relationships between the pressure-Hessian tensor and the vorticity/strain-rate tensor in the central region of the channel are consistent with that of isotropic turbulence. However, in the buffer layer with relatively stronger inhomogeneity and anisotropy, the vorticity tends to be aligned with the first or second eigenvector of the pressure-Hessian tensor in the unstable focus/compressing topological region, and tends to be aligned with the first eigenvector of the pressure-Hessian tensor in the stable focus/stretching topological region. In the unstable node/saddle/saddle and stable node/saddle/saddle topological regions, the vorticity prefers to lie in the plane of the first and second eigenvectors of the pressure-Hessian tensor. The strain-rate and the pressure-Hessian tensors tend to share their second principal direction. Moreover, for the coupling between the pressure-Hessian tensor and the principal strain rates, we clarify the influence on dissipation, the nonlinear generation of dissipation and the enstrophy generation. The decomposition of the pressure-Hessian tensor further shows that the slow pressure-related term dominates the pressure-Hessian tensor's contribution, and the influence of inhomogeneity and anisotropy mainly originates from the inhomogeneity and anisotropy of the fluctuating velocity. These statistical properties would be instructive in formulating dynamical models of the velocity gradient tensor for wall turbulence.

The Stokes equation describes the motion of fluids when inertial forces are negligible compared with viscous forces. In this article, we explore the consequence of parity-violating and non-dissipative (i.e. odd) viscosities on Stokes flows in three dimensions. Parity-violating viscosities are coefficients of the viscosity tensor that are not invariant under mirror reflections of space, while odd viscosities are those which do not contribute to dissipation of mechanical energy. These viscosities can occur in systems ranging from synthetic and biological active fluids to magnetized and rotating fluids. We first systematically enumerate all possible parity-violating viscosities compatible with cylindrical symmetry, highlighting their connection to potential microscopic realizations. Then, using a combination of analytical and numerical methods, we analyse the effects of parity-violating viscosities on the Stokeslet solution, on the flow past a sphere or a bubble and on many-particle sedimentation. In all the cases that we analyse, parity-violating viscosities give rise to an azimuthal flow even when the driving force is parallel to the axis of cylindrical symmetry. For a few sedimenting particles, the azimuthal flow bends the trajectories compared with a traditional Stokes flow. For a cloud of particles, the azimuthal flow impedes the transformation of the spherical cloud into a torus and the subsequent breakup into smaller parts that would otherwise occur. The presence of azimuthal flows in cylindrically symmetric systems (sphere, bubble, cloud of particles) can serve as a probe for parity-violating viscosities in experimental systems.

We propose a self-supervised cluster-based hierarchical reduced-order modelling methodology to model and analyse the complex dynamics arising from a sequence of bifurcations for a two-dimensional incompressible flow of the fluidic pinball. The hierarchy is guided by a triple decomposition separating a slowly varying base flow, dominant shedding and secondary flow structures. All these flow components are kinematically resolved by a hierarchy of clusters. The transition dynamics between these clusters is described by a directed network, called the cluster-based hierarchical network model (HiCNM). Three consecutive Reynolds number regimes for different dynamics are considered: (i) periodic shedding at $Re=80$, (ii) quasi-periodic shedding at $Re=105$ and (iii) chaotic shedding at $Re=130$, involving three unstable fixed points, three limit cycles, two quasi-periodic attractors and a chaotic attractor. The HiCNM enables identification of the dynamics between multiple invariant sets in a self-supervised manner. Both the global trends and the local structures during the transition are well resolved by a moderate number of hierarchical clusters. The proposed HiCNM provides a visual representation of transient and post-transient, multi-frequency, multi-attractor behaviour and may automate the identification and analysis of complex dynamics with multiple scales and multiple invariant sets.

Gravitational settling of a droplet in air onto a soft substrate is a ubiquitous event relevant to many natural processes and engineering applications. We study this phenomenon by developing a three-phase lubrication model of droplet settling onto a solid substrate coated by a thin soft layer represented by a viscous film, an elastic compressible layer and an elastic sheet supported by a viscous film. By combining scaling analysis, analytical methods and numerical simulations we elucidate how the resulting droplet dynamics is affected by the nature of the soft layer. We show that these soft layers can significantly affect the droplet shape during gravitational settling. When there is a linear response of the deformations of the soft layer, the air layer takes longer to drain as compared with the case of a droplet settling onto a rigid substrate. Our results provide new insight into the coupled interactions between droplets and solids coated by a thin film of a soft material.

We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate both the eigenfunctions to reveal the instability mechanism and the time-dependent states resulting from it, which provides a coherent picture of the phenomenon. The transition is of the saddle-node type, both with and without surfactant. The flow becomes unstable under stationary linear perturbations. Surfactant considerably reduces the interval of stable capillary numbers. Inertia increases the droplet deformation and decreases the critical capillary number. In the presence of the surfactant monolayer, neither the droplet deformation nor the stability is significantly affected by the droplet viscosity. The transient state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming occurs only in the presence of surfactants. The critical eigenmode leading to tip streaming is qualitatively the same as that yielding the central pinching mode for a clean interface, which indicates that the small local scale characterizing tip streaming is set during the nonlinear droplet deformation. The viscous surface stress does not significantly affect the droplet deformation and the critical capillary number. However, the damping rate of the dominant mode considerably decreases for viscous surfactants. Interestingly, shear viscous surface stress considerably alters the tip streaming arising in the supercritical regime, even for very small surface viscosities. The viscous surface stresses alter the balance of normal interfacial stresses and affect the surfactant transport over the stretched interface.

The water entry of a wedge has become a model test in marine and naval engineering research. Wagner theory, originating in 1932, predicts impact pressures, and accounts for contributions to the total pressure arising from various flow domains in the vicinity of the wetting region on the wedge. Here we study the slamming pressures on a wedge and a cone, impacting on water keeping a constant, well-controlled velocity, throughout the process, using high-fidelity sensors. Pressures at two locations on the impactor are measured during and after impact. Pressure time series from the two impactors are discussed using inertial pressure and time scales. The non-dimensionalised pressure time series are compared with sensor-integrated averaged composite Wagner solutions, Logvinovich solution (Hydrodynamics of Flows with Free Boundaries. Naukova Dumka, 1969, 4.7), modified Logvinovich solution, and generalised Wagner models. In addition, we provide an independent experimental justification of approximations made in the literature in extending the Wagner model to three dimensions. The second part of the paper deals with pre-impact air cushioning – an important concern since it is responsible for determining the thickness of the air layer trapped upon impact. Using a custom-made technique we measure the air–water interface dynamics as it responds to the build up of pressure in the air layer intervening in between the impactor and the free surface. We show both experimentally and using two-fluid boundary integral simulations, that the pre-impact deflection of the interface due to air-cushioning is fully described by potential flow.

We consider the stability of flux-driven flow through a long planar rigid channel, where a segment of one wall is replaced by a pre-tensioned hyperelastic (neo-Hookean) solid of finite thickness and subject to a uniform external pressure. We construct the steady configuration of the nonlinear system using Newton's method with spectral collocation and high-order finite differences. In agreement with previous studies, which use an asymptotically thin wall, we show that the thick-walled system always has at least one stable steady configuration, while for large Reynolds numbers the system exhibits three co-existing steady states for a range of external pressures. Two of these steady configurations are stable to non-oscillatory perturbations, one where the flexible wall is inflated (the upper branch) and one where the flexible wall is collapsed (the lower branch), connected by an unstable intermediate branch. We test the stability of these steady configurations to oscillatory perturbations using both a global eigensolver (constructed based on an analytical domain mapping technique) and also fully nonlinear simulations. We find that both the lower and upper branches of steady solutions can become unstable to self-excited oscillations, where the oscillating wall profile has two extrema. In the absence of wall inertia, increasing wall thickness partially stabilises the onset of oscillations, but the effect remains weak until the wall thickness becomes comparable to the width of the undeformed channel. However, with finite wall inertia and a relatively thick wall, higher-frequency modes of oscillation dominate the primary global instability for large Reynolds numbers.

Wind turbine wakes, being convectively unstable, may behave as an amplifier of upstream perturbations and make downstream turbines experience strong inflow fluctuations. In this work, we investigate the effects of the side-to-side motion of a floating offshore wind turbine (FOWT) on wake dynamics using large-eddy simulation and linear stability analysis (LSA). When the inflow turbulence intensity is low, simulation results reveal that the turbine motion with certain Strouhal numbers $St = fD/U_\infty \in (0.2,0.6)$ (where $f$ is the motion frequency, $D$ is the rotor diameter, and $U_\infty$ is the incoming wind speed), which overlap with the Strouhal numbers of wake meandering induced by the shear layer instability, can lead to wake meandering with amplitudes being one order of magnitude larger than the FOWT motion for the most unstable frequency. For high inflow turbulence intensity, on the other hand, the onset of wake meandering is dominated by the inflow turbulence. The probability density function of the spanwise instantaneous wake centres is observed being non-Gaussian and closely related to that of the side-to-side motion. This complements the existing wake meandering mechanisms, that the side-to-side motion of an FOWT can be a novel origin for the onset of wake meandering. It is also found that LSA can predict the least stable frequencies and the amplification factor with acceptable accuracy for motion amplitude $0.01D$. Effects of nonlinearity are observed when motion amplitude increases to $0.04D$, for which the most unstable turbine oscillations shift slightly to lower frequencies and the amplification factor decreases.

Large-eddy simulations (LES) are widely used for computing high Reynolds number turbulent flows. Spatial filtering theory for LES is not without its shortcomings, including how to define filtering for wall-bounded flows, commutation errors for non-uniform filters and extensibility to flows with additional complexity, such as multiphase flows. In this paper, the theory for LES is reimagined using a coarsening procedure that imitates nature. This physics-inspired coarsening approach is equivalent to Gaussian filtering for single-phase wall-free flows but opens up new insights for both physical understanding and modelling even in that simple case. For example, an alternative to the Germano identity is introduced and used to define a dynamic procedure without the need for a test filter. Non-uniform resolution can be represented in this framework without commutation errors, and the divergence-free condition is retained for incompressible flows. Potential extensions of the theory to more complex physics such as multiphase flows are briefly discussed.

Constant-flux gravity currents of viscoplastic fluid remain axisymmetric when extruded onto a dry horizontal plane. However, if the plane is coated with a shallow layer of water, the current suffers a dramatic non-axisymmetric instability in which localized $v$-shaped cuts appear in the outer edge where the viscoplastic fluid is in contact with water. These ‘fractures’ lengthen and guide the subsequent radial outflow, leading to distinctive flower-like patterns. This pattern formation process is illustrated for two viscoplastic materials, an aqueous suspension of Carbopol, and a mixture of water and joint compound (a kaolin-based, commercially available product). The fracturing spreads over the entire upper surface of the current when deeper water baths are used, complicating the extrusion patterns. The instability can be removed entirely when the ambient water layer is replaced by an immiscible liquid of comparable viscosity, indicating that the presence of water at the surface is key to the pattern formation process. We conjecture that the underlying mechanism is the fracture under tension of the viscoplastic material, exacerbated by the ambient water.

This study proposes a novel approach for building surrogate models, in the form of reduced-order models(ROMs), for partial differential equation constrained optimization. A physics-constrained data-driven framework is developed to transform large-scale nonlinear optimization problems into ROM-constrained optimization problems. Unlike conventional methods, which suffer from instability of the forward sensitivity function, the proposed approach maps optimization problems to system dynamics optimization problems in Hilbert space to improve stability, reduce memory requirements, and lower computational cost. The utility of this approach is demonstrated for aerodynamic optimization of an NACA 0012 airfoil at $Re = 1000$. A drag reduction of 9.35 % is obtained at an effective angle of attack of eight degrees, with negligible impact on lift. Similarly, a drag reduction of 20 % is obtained for fully separated flow at an angle of attack of $25^{\circ }$. Results from these two optimization problems also reveal relationships between optimization in physical space and optimization of dynamical behaviours in Hilbert space.

Internal gravity wave fields are decomposed into temporal modes revealing the hierarchical structure of nonlinear wave–wave interactions. We present a novel fusion of Green's functions for solving the forced internal wave equation with a weakly nonlinear perturbation expansion. Our approach is semi-analytical, based on integration over finite elements with the perturbation expansion ensuring source terms at each order are only dependent on the solutions at lower orders. Thus, the procedure is purely inductive and efficient to compute. To perform a thorough validation of our new method, we diagnose experiments using synthetic Schlieren and apply sophisticated post-processing techniques, including dynamic mode decomposition, to obtain these temporal modes for systems with discrete input frequencies. By decomposing the experimental field and comparing individual constituents against equivalents synthesised by our model, we are able to present the first truly comprehensive, validated, mechanistic picture of wave–wave interactions to arbitrary order. This synergy enables us to identify non-wave oscillatory behaviour at frequencies shared by waves in the hierarchy and leads us to discover an important open question regarding transmission efficiency within individual wave–wave interactions. Although our experiments are generated by boundary displacements, we present equivalences between source terms and boundary displacements so that the class of applicable systems may be broadened. Our technique also generalises to aperiodic and unbounded configurations and to any weakly nonlinear wave-governed system for which there is an available Green's function.

We study the development and the breaking process of standing waves at the interface between two miscible fluids of small density contrast. In our experiment, a subharmonic wave is generated by a time-periodic vertical acceleration via the Faraday instability. It is shown that its wavelength may be selected not only by the linear process predicted by the Floquet theory and favouring the most unstable modes allowed by the tank geometry, but also by a nonlinear mode competition mechanism giving the preference to subcritical modes. Subsequently, as the standing wave amplitude grows, a secondary destabilization process occurs at smaller scales and produces turbulent mixing at the nodes. We explain this phenomenon as a subcritical parametric resonance instability. Different approaches derived from local and global stability analysis are proposed to predict the critical wave steepness. These theories are then assessed against various numerical and experimental data varying the frequencies and amplitudes of the forcing acceleration.

A fully developed approach flow is necessary in open channel studies to maintain commonality among datasets obtained from different facilities. Two-component planar particle image velocimetry is used to study the characteristics of fully developed smooth open channel flow at a constant Reynolds number of 3.9 × 104 based on the maximum velocity and flow depth. The near-bed boundary layer is tripped to achieve a fully developed state and compared with the under- and over-tripped cases. The Reynolds stresses and higher-order moments are used as indicators to establish the fully developed state. Flow properties are explored by identifying uniform momentum zones (UMZs) using the probability density function of streamwise velocities. The instances are grouped based on the number of UMZs (NUMZ) and conditional averaging of flow variables of each group is used to evaluate the difference in flow properties between the developed and the developing flow. Large-scale ejections are found in the logarithmic layer when NUMZ is higher, whereas a lower number indicates the existence of large-scale sweeping motions. The distribution of the conditionally averaged ratio of the shear contribution from ejections and sweeps and velocity deficits shows a vertical variability in the fully developed state. The large-scale and pointwise quadrant events are used simultaneously to depict variability in inner flow properties between developing and fully developed flow which cannot be recognized in the mean flow characteristics. The sweep events have much higher shear generation in the outer flow in the fully developed state whereas the shear stress contribution from ejection is lower than that in developing flow.