This paper explores decaying turbulence beneath surface waves that is initially isotropic and shear free. We start by presenting phenomenology revealed by wave-averaged numerical simulations: an accumulation of angular momentum in coherent vortices perpendicular to the direction of wave propagation, suppression of kinetic energy dissipation and the development of depth-alternating jets. We interpret these features through an analogy with rotating turbulence (Holm 1996 Physica D. 98, 415–441), wherein the curl of the Stokes drift, ${\boldsymbol{\nabla}} \times {\boldsymbol{u^{S}}}$, takes on the role of the background vorticity (for example, $(f_0 + \beta y) {\boldsymbol{\hat{z}}}$ on the beta plane). We pursue this thread further by showing that a two-equation model proposed by Bardina et al. (1985 J. Fluid Mech. 154, 321–336) for rotating turbulence reproduces the simulated evolution of volume-integrated kinetic energy. This success of the two-equation model – which explicitly parametrises wave-driven suppression of kinetic energy dissipation – carries implications for modelling turbulent mixing in the ocean surface boundary layer. We conclude with a discussion about a wave-averaged analogue of the Rossby number appearing in the two-equation model, which we term the ‘pseudovorticity number’ after the pseudovorticity ${\boldsymbol{\nabla }} \times {\boldsymbol{u}}^S$. The pseudovorticity number is related to the Langmuir number in an integral sense.

Compliant walls made from homogeneous viscoelastic materials may attenuate the amplification of Tollmien–Schlichting waves (TSWs) in a two-dimensional boundary-layer flow, but they also amplify travelling-wave flutter (TWF) instabilities at the interface between the fluid and the solid, which may lead to a premature laminar-to-turbulent transition. To mitigate the detrimental amplification of TWF, we propose to design compliant surfaces using phononic structures that aim at avoiding the propagation of elastic waves in the solid in the frequency range corresponding to the TWF. Thus, stiff inserts are periodically incorporated into the viscoelastic wall in order to create a band gap in the frequency spectrum of the purely solid modes. Fluid–structural resolvent analysis shows that a significant reduction in the amplification peak related to TWF is achieved while only marginal deterioration in the control of TSWs is observed. This observation suggests that the control of TSWs is still achieved by the overall compliance of the wall, while the periodic inserts inhibit the amplification of TWF. Bloch analysis is employed to discuss the propagation of elastic waves in the phononic surface to deduce design principles, accounting for the interaction with the flow.

Large numbers of relative periodic orbits (RPOs) have been found recently in doubly periodic, two-dimensional Kolmogorov flow at moderate Reynolds numbers ${\textit{Re}} \in \{40, 100\}$. While these solutions lead to robust statistical reconstructions at the ${\textit{Re}}$ values where they were obtained, it is unclear how their dynamical importance changes with ${\textit{Re}}$. Arclength continuation on this library of solutions reveals that large numbers of RPOs quickly become dynamically irrelevant, reaching dissipation values either much larger or smaller than the values typical of the turbulent attractor at high ${\textit{Re}}$. The scaling of the high-dissipation RPOs is shown to be consistent with a direct connection to solutions of the unforced Euler equation, and is observed for a wide variety of states beyond the ‘unimodal’ solutions considered in previous work (Kim & Okamoto, Nonlinearity vol. 28, 2015, p. 3219). However, the weakly dissipative states have properties indicating a connection to exact solutions of a forced Euler equation. The dynamical irrelevance of many solutions leads to poor statistical reconstruction at higher ${\textit{Re}}$, raising serious questions for the future use of RPOs for estimating probability densities. Motivated by the Euler connection of some of our RPOs, we also show that many of these states can be well described by exact relative periodic solutions in a system of point vortices. The point vortex RPOs are converged via gradient-based optimisation of a scalar loss function which (i) matches the dynamics of the point vortices to the turbulent vortex cores and (ii) insists the point vortex evolution is itself time-periodic.

In the present study, we observe interesting profiles and fluctuations in a quasi-two-dimensional thermal convection system filled with low-Prandtl-number liquid metal. A high-precision thermistor, which can be precisely controlled to move up and down, is used to measure the temperature distribution along the centreline of a convection cell. As the thermistor probes move away from the heated wall surface, the measured temperatures initially decrease to values below the central temperature of the cell, then recover to the central temperature, indicating an inverse temperature gradient. Furthermore, by analysing the root-mean-square temperature ($\sigma _T (z)$) along the centreline, we find a second peak away from the wall location, which has never been reported before, in addition to the first peak associated with the thermal boundary thickness. This phenomenon is also confirmed by the results of third- and fourth-order moments of temperature. Experimental results, together with insights from previous studies, suggest that in liquid metal, the distinct flow organisation arising from the large thermal diffusivity plays an important role in shaping the observed temperature distribution.

Coastal areas face unprecedented challenges from accelerating sea-level rise, increasing urbanisation and biodiversity loss, necessitating sustainable coastal protection strategies that go beyond traditional engineering approaches: While Nature-based Solutions (NbS) offer ecological benefits, their implementation faces constraints in space, timely readiness and standardisation. Hybrid Nature-based Solutions (HNbS) have emerged as promising alternatives, yet current taxonomic classifications remain ambiguous and insufficient to accommodate emerging technologies. This synthetic review analyses the evolution and current role of NbS in coastal climate change adaptation through a comprehensive juxtaposition of design principles and functional mechanisms of engineered and nature-based coastal defences. The review synthesises knowledge from sustainable climate adaptation and digital fabrication literature to establish precise taxonomic classifications for solutions that integrate engineered and nature-based approaches, namely HNbS. The analysis reveals gaps in the existing HNbS taxonomy, particularly regarding structures enabled by digital fabrication technologies. The three identified, distinct categories of HNbS are: (1) Hybrid Nature-based Strategies, combining engineered and natural elements at planning scales; (2) Hybrid Nature-based Modules, integrating both components within individual structures; and (3) Confluent Hybrid Nature-based Solutions, representing an emerging category where engineering and natural systems converge at material or microorganism scales, offering distinctly engineered infrastructures with natural characteristics. While contemporary hybrid approaches are being implemented, Confluent Hybrid Nature-based Solutions under research, may face a critical timing mismatch due to a gap between lengthy innovation timelines and urgent adaptation needs. Dynamic Adaptive Policy Pathways provide a framework for strategically accommodating these emerging innovations, enabling early-stage solutions and performance demonstration under real-world conditions. The new taxonomic framework outlined in this study prevents imprecise terminology and provides a foundation for robust, low-regret coastal adaptation strategies addressing contemporary and future coastal pressures like climate change impacts and biodiversity conservation requirements.

Pendant drops appear in many engineering applications, such as inkjet printing and optical tensiometry, and they have also been the subject of studies of droplet–particle interaction. While the hydrostatics of pendant drops has been studied extensively, the influence of external flow disturbances has received limited attention. This research aims to incorporate aerodynamic factors into the understanding of pendant drop behaviour. Employing a simplified model, an irrotational flow aligned with the drop’s axis is derived from a distribution of singularity elements within the drop. The drop’s equilibrium shape is then determined using a numerical model that couples the flow field with the Young–Laplace equation. The model’s predictions are compared to droplet images captured via high-speed shadowgraph in a vertical wind tunnel, showing good agreement with the experimentally observed shapes. Additionally, under certain flow conditions, the drop exhibits instability in the form of periodic pendulum-like motion. This instability was linked to two distinct critical drop heights, and the corresponding stability criterion was mathematically derived from the numerical model. Our theoretical and experimental findings provide the first quantitative description of the equilibrium shape and stability criterion of pendant drops under the influence of external flow.

Experiments have shown that ultrasound-stimulated microbubbles can translate through gel phantoms and tissues, leaving behind tunnel-like degraded regions. A computational model is used to examine the tunnelling mechanisms in a model material with well-defined properties. The high strain rates motivate the neglect of weak elasticity in favour of viscosity, which is taken to degrade above a strain threshold. The reference parameters are motivated by a 1 $\unicode{x03BC}$m diameter bubble in a polysaccharide gel tissue phantom. This is a reduced model and data are scarce, so close quantitative agreement is not expected, but tunnels matching observations do form at realistic rates, which provides validation sufficient to analyse potential mechanisms. Simulations of up to 100 acoustic cycles are used to track tunnelling over 10 bubble diameters, including a steady tunnelling phase during which tunnels extend each forcing cycle in two steps: strain degrades the tunnel front during the bubble expansion, and then the bubble is drawn further along the tunnel during its subsequent inertial collapse. Bubble collapse jetting is damaging, though it is only observed during a transient for some initial conditions. There is a threshold behaviour when the viscosity of the undamaged material changes the character of the inertial bubble oscillation. Apart from that, the tunnel growth rate is relatively insensitive to the high viscosity of the material. Higher excitation amplitudes and lower frequencies accelerate tunnelling. That acoustic radiation force, elasticity and bubble jetting are not required is a principal conclusion.

An asymptotic model for the flow of a highly viscous film coating the interior of a slippery, flexible tube is developed and studied. The model is valid for the axisymmetric flow of moderately thick films, and accounts for tube flexibility, wall damping, longitudinal tension, slip length and strength of base flow due either to gravity or airflow. In the absence of base flow, linear stability analysis shows the existence of one unstable mode; the presence of base flow allows for multiple unstable modes arising due to the Plateau–Rayleigh instability and elastic instability, with stronger base flow reducing the maximum growth rate. Numerical solutions in the absence of base flow show that slip decreases the amplitude of wall deformations and can significantly decrease the time to plug formation in weakly flexible or strongly damped tubes. For falling films, the impact of model parameters on the critical thickness required for plug formation was analysed by studying turning points in families of travelling-wave solutions; this thickness decreases with slip, flexibility and tension, while damping had a non-monotonic impact on critical thickness. In contrast to model solutions in rigid tubes, for flexible tubes the critical thickness cannot be made arbitrarily large through simply increasing the strength of the base flow. For air-driven films, both slip and flexibility increase the rate of film transport along the tube.

We investigate theoretically the breakup dynamics of an elasto-visco-plastic filament surrounded by an inert gas. The filament is initially placed between two coaxial disks, and the upper disk is suddenly pulled away, inducing deformation due to both constant stretching and capillary forces. We model the rheological response of the material with the Saramito–Herschel–Bulkley (SHB) model. Assuming axial symmetry, the mass and momentum balance equations, along with the constitutive equation, are solved using the finite element framework PEGAFEM-V, enhanced with adaptive mesh refinement with an underlying elliptic mesh generation algorithm. As the minimum radius decreases, the breakup dynamics accelerates significantly. We demonstrate that the evolution of the minimum radius, velocity and axial stress follow a power-law scaling, with the corresponding exponent depending on the SHB shear-thinning parameter, $n$. The scaling exponents obtained from our axisymmetric simulations under creeping flow are verified through asymptotic analysis of the slender filament equations. Our findings reveal three distinct breakup regimes: (a) elasto-plastic, (b) elasto-plasto-capillary, both with finite-time breakup for $n\lt 1$, and (c) elasto-plasto-capillary with no finite-time breakup for $n=1$. We show that self-similar solutions close to filament breakup can be achieved by appropriate rescaling of length, velocity and stress. Notably, the effect of the yield stress becomes negligible in the late stages of breakup due to the local dominance of high elastic stresses. Moreover, the scaling exponents are independent of elasticity, resembling the breakup behaviour of finite extensible viscoelastic materials.

In air-entraining flows, there is often strong turbulence beneath the free surface. We consider the entrainment of bubbles at the free surface by this strong free-surface turbulence (FST). Our interest is the entrainment size distribution (per unit free surface area) $I(a)/A_{\textit{FS}}$, for bubbles with radius $a$ greater than the capillary scale ($\approx 1.3\ \mathrm{mm}$ for air–water on Earth), where gravity dominates surface tension. We develop a mechanistic model based on entrained bubble size being proportional to the minimum radius of curvature of the initial surface deformation. Using direct numerical simulation of a flow that isolates entrainment by FST, we show that, consistent with our mechanism, $I(a)/A_{\textit{FS}} = C_I \, g^{-3} \varepsilon ^{7/3} (2 a)^{-14/3}$, where $g$ is gravity, and $\varepsilon$ is the turbulence dissipation rate. In the limit of negligible surface tension, $C_I\approx 3.62$, and we describe how $C_I$ decreases with increasing surface tension. This scaling holds for sufficiently strong FST such that near-surface turbulence is nearly isotropic, which we show is true for turbulent Froude number ${\textit{Fr}}^2_T = \varepsilon /u_{\textit{rms}} g \gt 0.1$. While we study FST entrainment in isolation, our model corroborates previous numerical results from shear-driven flow, and experimental results from open-channel flow, showing that the FST entrainment mechanism that we elucidate can be important in broad classes of air-entraining flows.

The propagation of linear waves in non-ideal compressible fluids plays a crucial role in numerous physical and engineering applications, particularly in the study of instabilities, aeroacoustics and turbulence modelling. This work investigates linear waves in viscous and heat-conducting non-ideal compressible fluids, modelled by the Navier–Stokes–Fourier equations and a fully arbitrary equation of state (EOS). The linearised governing equations are derived to analyse the dispersion relations when the EOS differs from that of an ideal gas. Special attention is given to the influence of non-ideal effects and various dimensionless numbers on wave propagation speed and attenuation. By extending classical results from Kovásznay (1953 J. Aeronaut. Sci. vol. 20, no. 10, pp. 657–674) and Chu (1965 Acta Mech. vol. 1, no. 3, pp. 215–234) obtained under the ideal gas assumption, this study highlights the modifications introduced by arbitrary EOSs to the linear wave dynamics in non-ideal compressible flows. This work paves the path for an improved understanding and modelling of wave propagation, turbulence and linear stability in arbitrary viscous and heat-conducting fluids.

The complex behaviour of air–liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder et al. (1986, Phys. Rev. A, vol. 34, 5175) to present a systematic investigation of the dynamics that results when a small air bubble is placed at the tip of a steadily propagating air finger in a horizontal Hele-Shaw channel. The system can exhibit steady and oscillatory behaviours, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger’s tip. At fixed flow rate, the oscillations commence when the bubble size is increased sufficiently that the decreased curvature of the bubble tip in the horizontal plane matches that of the finger tip. This causes the axial pressure gradient along the bubble to vanish, thus rendering the bubble susceptible to lateral perturbations. Differing time scales for finger and bubble restoral allow sustained oscillations to develop in the finger–bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a single finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie disorder in viscous fingering.

Small-scale topography can significantly influence large-scale motions in geophysical flows, but the dominant mechanisms underlying this complicated process are poorly understood. Here, we present a systematic experimental study of the effect of small-scale topography on zonal jets. The jet flows form under the conditions of fast rotation, a uniform background $\beta$-effect, and sink–source forcing. The small-scale topography is produced by attaching numerous small cones on the curved bottom plate, and the height of the cones is much smaller than the water depth. It is found that for all tested cases, the energy fraction in the zonal mean flow consistently follows a scaling $E_{uZ}/E_{uT}=C_1 l_f^2\epsilon _{\textit{up}}^{-2/5}\beta _{\textit{eff}}^{6/5}$, where $l_f$ is the forcing scale, $\epsilon _{\textit{up}}$ is the upscale energy transfer rate, and $\beta _{\textit{eff}}$ measures the effective $\beta$-effect in the presence of topography. The presence of the small-scale topography weakens the jet strength notably. Moreover, the effect of topography on energy transfers depends on the topography magnitude $\beta _\eta$, and there exist three regimes. At small $\beta _\eta$, the inverse energy transfers are remarkably diminished while the jet pattern remains unchanged. When $\beta _\eta$ increases, a blocked flow pattern forms, and the jet width reaches saturation, becoming independent of the forcing magnitude and $\beta$. At moderate $\beta _\eta$, the inverse energy fluxes are surprisingly enhanced. A further increase of $\beta _\eta$ leads to a greater reduction of the energy fluxes. We finally examine the effect of topography from the perspective of turbulence–topography interaction.

We investigate interactions between two like-signed vortices over either an isolated seamount or a basin (a depression in the bathymetry), using a quasi-geostrophic, two-layer model on the $f$-plane. When the vortex pair is centred over the seamount, the vortices are pushed together by the secondary flow generated in the bottom layer, facilitating their merger. Over a basin, the deep anomalies are much stronger and their interaction strains out the surface vortices. The results are supported by an analytical estimation of the initial potential vorticity anomalies in the lower layer and by analysis of the linear stability of a single vortex over the bathymetry. Similar phenomena are observed when the vortex pair is displaced from the bathymetric centre and when the initial vortices are initially compensated. Sub-deformation-scale vortices are less influenced by bathymetry than larger vortices. The results help explain asymmetries noted previously in turbulence simulations over bathymetry.