When surface waves interact with ambient turbulence, the two affect each other mutually. Turbulent eddies get redirected, intensified and periodically stretched and compressed, while the waves suffer directional scattering. We study these mutual interactions experimentally in the water channel laboratory at the Norwegian University of Science and Technology (NTNU) Trondheim. Long groups of waves were propagated upstream on currents with identical mean flow but different turbulence properties, created by an active grid at the current inlet. The subsurface flow in the spanwise–vertical plane was measured with stereo particle-image velocimetry. Comparing the subsurface velocity fields before and after the passage of a wave group, a strong enhancement of streamwise vorticity is observed which increases rapidly towards the surface for $k_0z\gtrsim -0.3$ ($z$, vertical distance from still surface; $k_0$, carrier wavenumber) in qualitative agreement with theory. Next, we measure the broadening of the directional wave spectrum at increasing propagation distance. The rate of directional diffusion is greatest for the turbulent case with the highest energy at the longest length scales whereas the highest total turbulent kinetic energy overall did not produce the most scattering. The variance of directional spectra is found to increase linearly as a function of propagation time.

Despite elegant theoretical descriptions and numerous potential applications in nature, the various features of wave turbulence and its associated transfers across scales are still debated. One reason is the difficulty, both experimentally and numerically, of studying a chaotic nonlinear wavy system with sufficient precision, over a sufficient duration, and with sufficient separation between the injection scale, the system scale and the dissipation scale, to dwell in detail on the description of a statistically converged state. The numerical study of Thomas & Ding (J. Fluid Mech., 2023, this issue) sheds new light on one key aspect of this global problem: the upscale transfer of waves in rotating shallow water equations, which is of relevance to atmosphere and ocean dynamics. Their results first confirm the theoretical predictions, with a robust inverse wave cascade, a predominant role of quartic resonances and a slope value of the wave spectrum in agreement with the expected range. But their deeper analysis of the results also highlights some weakness in the theoretical foundations, exhibiting strong intermittency and departure from ideal Gaussian statistics. This work thus calls for improved modelling, acknowledging that important large-scale climatic features could actually result from the piling up of small-scale waves, unresolved by typical Earth system models. Beyond rotating shallow water systems, this study more generically resonates with open questions in the field of wave turbulence and its geophysical applications, including recent research in deep rotating and stably stratified systems.

The interaction of a hollow droplet impacting a solid surface occurs in several applications, including controllable biomedicine and thermal spray coating. Understanding the physics of the hollow droplet spreading is the key to maintaining the mass transfer process in all relevant applications. In this work, a comprehensive experimental, numerical and theoretical study is performed on water hollow droplets impacting a rigid surface to better understand the flattening process of a hollow droplet. In the numerical part, compressible Navier–Stokes equations are solved using the volume of fluid (VOF) method in a two-dimensional (2-D)-axisymmetric model. The comparison of simulation results with the experimental photographs shows that the numerical solution can correctly predict the hollow droplet shape evolution. The results show that the spreading diameter and height of the counter-jet formed after the hollow droplet impact grow with impact velocity. Investigating the size and location of the entrapped bubble shows an optimum bubble size that facilitates the hollow droplet flattening. It is also shown that the ripples on splats produced by the hollow droplets with a larger bubble size are higher than those of small bubbles. In the end, a theoretical model is developed to analyse the maximum spreading diameter of the hollow droplet impact analytically. Its prediction is in good agreement with the experimental and numerical results.

We investigate how the heat flux $Nu$ scales with the imposed temperature gradient $Ra$ in homogeneous Rayleigh–Bénard convection using one-, two- and three-dimensional simulations on logarithmic lattices. Logarithmic lattices are a new spectral decimation framework which enables us to span an unprecedented range of parameters ($Ra$, $Re$, $\Pr$) and test existing theories using little computational power. We first show that known diverging solutions can be suppressed with a large-scale friction. In the turbulent regime, for $\Pr \approx 1$, the heat flux becomes independent of viscous processes (‘asymptotic ultimate regime’, $Nu\sim Ra ^{1/2}$ with no logarithmic correction). We recover scalings coherent with the theory developed by Grossmann and Lohse, for all situations where the large-scale frictions keep a constant magnitude with respect to viscous and diffusive dissipation. We also identify another turbulent friction-dominated regime at $\Pr \ll 1$, where deviations from the Grossmann and Lohse prediction are observed. These two friction-dominated regimes may be relevant in some geophysical or astrophysical situations, where large-scale friction arises due to rotation, stratification or magnetic field.

We experimentally investigate the settling of millimetre-sized thin disks in quiescent air. The range of physical parameters is chosen to be relevant to plate crystals settling in the atmosphere: the diameter-to-thickness aspect ratio is $\chi =25\unicode{x2013}60$, the Reynolds numbers based on the disk diameter and fall speed are $Re=O(10^2)$ and the inertia ratio is $I^*=O(1)$. Thousands of trajectories are reconstructed for each disk type by planar high-speed imaging, using the method developed by Baker & Coletti (J. Fluid Mech., vol. 943, 2022, A27). Most disks either fall straight vertically with their maximum projected area normal to gravity or tumble while drifting laterally at an angle $<20^\circ$. Two of the three disk sizes considered exhibit bimodal behaviour, with both non-tumbling and tumbling modes occurring with significant probabilities, which stresses the need for a statistical characterization of the process. The smaller disks (1 mm in diameter, $Re=96$) have a stronger tendency to tumble than the larger disks (3 mm in diameter, $Re=360$), at odds with the diffused notion that $Re=100$ is a threshold below which falling disks remain horizontal. Larger fall speeds (and, thus, smaller drag coefficients) are found with respect to existing correlations based on experiments in liquids, demonstrating the role of the density ratio in setting the vertical velocity. The data supports a simple scaling of the rotational frequency based on the equilibrium between drag and gravity, which remains to be tested in further studies where disk thickness and density ratio are varied.

In this paper, the possibility of absolute instability in a plane unidirectional jet is analysed. We consider a parametrized family of velocity profiles with variable inflection point location and shear layer thickness. Using the inviscid saddle-point analysis, we show that absolute instability can occur in the case of a sufficiently low velocity at the inflection point or a sufficiently thin shear layer. Then we proceed to the viscous analysis and find the critical Reynolds numbers separating the zones of convective and absolute instability. We obtained a minimum value $Re = 315$. As an independent verification of the theoretical results, we conduct a direct numerical simulation of the evolution of a localized pulse perturbation in the framework of the linearized Navier–Stokes equations. The calculated absolute/convective instability boundary is in a good agreement with theoretical results of the saddle-point analysis.

Zonal jet formation in $\beta$-plane turbulence is investigated with the focus on whether an accurate closure can be developed for the eddy momentum fluxes due to small-scale random forcing. The approach of Srinivasan and Young (J. Atmos. Sci., vol. 71, 2014, pp. 2169–2185) is developed to give a relatively simple expression for the local Reynolds stress, due to a white-in-time random forcing with characteristic length scale much less than the jet spacing. In typical jet flows, however, it is demonstrated that the Srinivasan–Young flux is not the full story because momentum fluxes due to jet-scale waves, present as a result of distinct barotropic instabilities of the eastward and westward jets, respectively, also play a key role in the momentum balance. Numerical simulations that explicitly include the random forcing are then compared with those in which the Srinivasan–Young closure is applied. For typical jet flows, good agreement of the equilibrium zonal flow is found provided that the closure simulation is not truncated to be purely zonal, i.e. jet-scale secondary barotropic instabilities are allowed to develop. Flows in which the geometry or external forcing acts to suppress the development of secondary instabilities are also simulated, and for these flows the Srinivasan–Young closure is shown to be successful as a purely zonal closure. It is argued that vortex condensates in isotropic forced-dissipative 2D turbulence are an example of this latter situation.

When liquid metal batteries are charged or discharged, strong electrical currents are passing through the three liquid layers that we find in their interior. This may result in the metal pad roll instability that drives gravity waves on the interfaces between the layers. In this paper, we investigate theoretically metal pad roll instability in idealised cylindrical liquid metal batteries that were simulated previously by Weber et al. (Phys. Fluids, vol. 29, no. 5, 2017b, 054101) and Horstmann et al. (J. Fluid Mech., vol. 845, 2018, pp. 1–35). Near the instability threshold, we expect weakly destabilised gravity waves, and in this parameter regime, we can use perturbation methods to find explicit formulas for the growth rate of all possible waves. This perturbative approach also allows us to include dissipative effects, hence we can locate the instability threshold with good precision. We show that our theoretical growth rates are in quantitative agreement with previous and new direct numerical simulations. We explain how our theory can be used to estimate a lower bound on cell size beneath which metal pad roll instability is unlikely.

In this work, the linear responses of turbulent mean flow to both harmonic and stochastic forcing are investigated for supersonic channel flow. Well-established universal relations are utilized to obtain efficiently the mean profiles with a large parameter space, with the bulk Mach number up to 5 and the friction Reynolds number up to $10^4$, so a systematic parameter study is feasible. The most amplified structure takes the form of streamwise velocity and temperature streaks forced optimally by the streamwise vortices. The outer peak of the pre-multiplied energy amplification corresponds to the large-scale motion, whose spanwise wavelength ($\lambda _z^+$) is very insensitive to compressibility effects. In contrast, the classic inner peak representing small-scale near-wall motions disappears for the stochastic response with increasing Mach number. Meanwhile, the small-scale motions become much less coherent. A decomposition of the forcing identifies different effects of the incompressible counterpart and the thermodynamic components. Wall-cooling effects, arising with high Mach number, increase the spacing of the most amplified near-wall streaks; the spacing becomes nearly invariant with Mach number if expressed in semi-local units. Meanwhile, the coherence of stochastic response with $\lambda _z^+>90$ is enhanced, but on the other hand, with $\lambda _z^+<90$ it is decreased. The geometrical self-similarity of the response in the mid-$\lambda _z$ range is still roughly satisfied, insensitive to Mach number. Finally, theoretical analyses of the perturbation equations are presented to help with understanding the scaling of energy amplification.

Supercritical ${\rm CO}_2$ injection and dissolution into deep brine aquifers allow its sequestration within geological formations. After injection, ${\rm CO}_{2}$ gas phase is buoyancy-driven over the denser aqueous brine, reaching an apparent gravitational stable distribution. However, ${\rm CO}_2$ dissolution in brine propels convection since the mixture is even denser than the underlying brine. This process still needs to be characterised comprehensively. Here, we investigate the irreversible mixing of dissolved ${\rm CO}_2$ in brine through laboratory-scale numerical experiments utilising the Hele-Shaw model (Letelier et al., J. Fluid Mech., vol. 864, 2019, pp. 746–767) and a fully miscible two-fluid system. In this scenario, mixing the less dense fluid – mimicking ${\rm CO}_{2}$ gas phase – with the heavier fluid – representing aqueous brine – catalyses cabbeling-powered convection. Our numerical simulations recover the laboratory results in porous media by Neufeld et al. (Geophys. Res. Lett., vol. 37, issue 22, 2010, L22404) and may explain the scaling law obtained by Backhaus et al. (Phys. Rev. Lett., vol. 106, issue 10, 2011, 104501) in Hele-Shaw cells. More remarkably, we show that the mass flux between the two analogue fluids, characterised by the Sherwood number $ {{Sh}}$, obeys the universal scaling law $ {{Sh}}\sim {{Ra}}\, \vartheta _{scalar}$, with $ {{Ra}}$ the Rayleigh number and $\vartheta _{scalar}$ the mean scalar dissipation rate. This paper sheds light on the fluid dynamics and solubility trapping in geological carbon sequestration.

This paper studies the sustainability of plastrons on superhydrophobic (SHPo) surfaces made of longitudinal micro-trenches covered by nano-grass with the main interest on hydrodynamic friction drag reduction in high-speed flows of open water, which represent the operating conditions of common watercraft. After revising the shear-driven drainage model to address the air diffusion for SHPo surfaces, the existing theories are combined to reveal the trends of how the immersion depth, air saturation level and shear stress affect the maximum attainable plastron length. Deviations from the theories by the dynamic effect at the two ends of the trench, the interfacial contaminations and turbulent fluctuation are also discussed. A combinatorial series of well-defined SHPo trench surfaces (4 cm × 7 cm in size with varying trench widths, depths, lengths and roughnesses) is microfabricated and attached underneath a 4 m long motorboat on seawater in turbulent flows up to 7.2 m s−1 (shear rate ∼83 000 s−1 and friction Reynolds number ∼5500). Because the plastron can provide a substantial slip only while its air–water interfaces are pinned (or only slightly depinned) at the trench top, two underwater cameras are employed to differentiate the pinned (and slightly depinned) interfaces from the depinned (and no) interfaces. In addition to achieving pinned plastrons on 6 cm long trenches aligned to high-speed flows in open water, the experimental results corroborate the theoretical estimations, supporting the design of SHPo surfaces for field applications.

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$. Three types of $\alpha$–$Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$. As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$. This implies that the ‘walking’ drop instability reported in previous literature does not exist.

The collapse of a gas or vapour bubble near a non-porous boundary is directed at the boundary due to the asymmetry induced by the nearby boundary. High surface pressure and shear stress from this collapse can damage, or clean, the surface. A porous boundary, such as a filter, would act similarly to a non-porous boundary but with reduced asymmetry and thus reduced effect. Prior research has measured the cleaning effect of bubbles on filters using ultrasonic cleaning, but it is not known how the bubble dynamics are fundamentally affected by the porosity of the surface. We address this question experimentally by investigating how the standoff distance, porosity, pore size and pore shape affect two collapse properties: bubble displacement and bubble rebound size. We show that these properties depend primarily on the standoff distance and porosity of the boundary and extend a previously developed numerical model that approximates this behaviour. Using the numerical model in combination with experimental data, we show that bubble displacement and bubble rebound size each collapse onto respective single curves.

We consider the stability of the drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape that evolves due to axial drawing, inertia and surface tension effects. Despite the complicated geometry of the boundaries, we use asymptotic techniques to determine a particularly convenient formulation of the equations of motion that is well-suited to stability calculations. We will determine an explicit asymptotic solution for steady states with (a) large surface tension and negligible inertia, and (b) large inertia. In both cases, we will show that complicated boundary layer structures can occur. We will use linear stability analysis to show that the presence of an axisymmetric hole destabilises the flow for finite capillary number and which answers a question raised in the literature. However, our formulation allows us to go much further and consider arbitrary hole structures or non-axisymmetric shapes, and show that any structure with holes will be less stable than the case of a solid axisymmetric thread. For a solid axisymmetric thread, we will also determine a closed-form expression that delineates the unconditional instability boundary in which case the thread is unstable for all draw ratios. We will determine how the detailed effects of the microstructure affect the stability, and show that they manifest themselves only via a single function that occurs in the stability problem and hence have a surprisingly limited effect on the stability.

This study investigates the effect of structural damping on vortex-induced vibration (VIV) of a circular cylinder when the mass ratio is below its critical value. It is confirmed by water-channel experiments and a reduced-order model (ROM) that the previously identified phenomenon of VIV forever, i.e. resonance oscillations at any reduced velocity, persists even with high structural damping. Of interest, the ROM results reveal that the wake mode for VIV forever is unstable with a constant positive growth rate with increasing reduced velocity, while the experimental results suggest that VIV forever is associated with a synchronisation between the non-stationary cylinder vibration frequency and the vortex-shedding frequency.

When motile algal cells are exposed to gyrotactic torques, their swimming directions are guided to form radial accumulation, well known as hydrodynamic focusing. The origin of hydrodynamic focusing from the effects of active swimming, ambient flow and particle anisotropy is elucidated in the present study on the pre-asymptotic dispersion of active particles through a vertical pipe. With an extension of the Galerkin method to pipe flows, time-dependent solutions directly from the Smoluchowski equation in the position and orientation space are derived by series expansions of spherical harmonics and Bessel functions. Ballistic and diffusive scaling laws are examined with the predominance of self-propelled swimming, and computation is validated against an explicit benchmark solution and Lagrangian particle simulation. In the limit of extreme shear, the competitive roles of shear dispersion and Brownian rotation are reflected concretely in the pre-asymptotic phase of hydrodynamic focusing. For flows with various shear strengths, a concentration peak in near-wall regions with a smooth transition to hydrodynamic focusing is illustrated with richer phenomena in upwelling and downwelling flows. A newly observed regime through a vertical pipe, named transient effective trapping, is revealed as a transitional mode towards hydrodynamic focusing. The pre-asymptotic approach to hydrodynamic focusing is elaborated intensively through extensive solutions of concentration moments and macroscopic transport coefficients characterised by swimming and flow Péclet numbers. The unique findings for the origin of hydrodynamic focusing could provide insight into related micro-algae reactor technology and contribute to flow control and biomass transfer in confined environments.

Microbubble emission boiling (MEB), a phenomenon that occurs under highly subcooled boiling conditions, can achieve a high heat flux beyond critical heat flux. MEB has been observed to differ from nucleate boiling and is always accompanied by the emission of microbubbles from an oscillating bubble. The heat transfer mechanism of MEB differs from that of nucleate boiling and remains elusive. In this study, we measure the behaviour of a vapour bubble and its induced liquid flow simultaneously during subcooled boiling using the high-speed two-phase particle image velocimetry method. Different from nucleate boiling, we observe a rapid oscillating flow outside the bubble that constantly attaches on the heating surface in MEB. The spatially and temporally averaged velocity magnitude values under this oscillating flow, and their dependency on liquid subcooling and heat flux, are quantified. Then we derive a scaling law for the heat transfer of MEB as a function of Péclet and Jacob numbers. With such correlations, we suggest that the oscillating flow induced by bubble oscillations is important to MEB, thus elucidating a different heat transport process from nucleate boiling.

Turbulence in the restricted nonlinear (RNL) dynamics is analysed and compared with direct numerical simulations (DNS) of Poiseuille turbulence at Reynolds number $R=1650$. The structures are obtained by proper orthogonal decomposition (POD) analysis of the two components of the flow partition used in RNL dynamics: the streamwise mean flow and fluctuations. POD analysis of the streamwise mean flow indicates that the dominant POD modes, in both DNS and RNL dynamics, are roll-streaks harmonic in the spanwise direction. However, we conclude that these POD modes do not occur in isolation but rather are Fourier components of a coherent roll-streak structure. POD analysis of the fluctuations in DNS and RNL dynamics reveals similar complex structures consisting in part of oblique waves collocated with the streak. The origin of these structures is identified by their correspondence to POD modes predicted using a stochastic turbulence model (STM). These predicted POD modes are dominated by the optimally growing structures on the streak, which the STM predicts correctly to be of sinuous oblique wave structure. This close correspondence between the roll-streak structure and the associated fluctuations in DNS, RNL dynamics and the STM implies that the self-sustaining mechanism operating in DNS is essentially the same as that in RNL dynamics, which has been associated previously with optimal perturbation growth on the streak.

Numerical simulations of the interaction of internal solitary waves (ISWs) of opposite polarity are conducted by solving the incompressible Euler equations under the Boussinesq approximation. A double-pycnocline stratification is used. A method to determine when ISWs of both polarities exist is also presented. The simulations confirm previous work that the interaction of waves of the same polarity are soliton-like; however, here it is shown that when a fast ISW with the same polarity as a Korteweg–de Vries (KdV) solitary wave catches up and interacts with a slower ISW of opposite polarity, the interaction can be far from soliton-like. The energy in the fast KdV-polarity wave can increase by more than a factor of 5 while the energy in the slower negative-KdV-polarity wave can decrease by 50 %. Large trailing wave trains may be generated and in some cases multiple ISWs with KdV polarity may be formed by the interaction.

Helmholtz decomposition of velocity perturbations is performed in conjunction with linear stability analysis to examine the effects of flow-thermodynamics interactions on the stability of high-speed boundary layers. A corresponding decomposition of the pressure field is also proposed. The contributions of perturbation solenoidal kinetic, dilatational kinetic and internal energy to the various instability modes are examined as a function of Mach number ($M$). As expected, dilatational and pressure field effects play an insignificant part in the first-mode behaviour at all Mach numbers. The second (Mack) mode, however, is dominantly dilatational in nature, and perturbation internal energy is significant compared to perturbation kinetic energy. The observed behaviour is explicated by examining the key processes of production and pressure-dilatation. Production of the second-mode dilatational kinetic energy is mostly due to the solenoidal-dilatational covariance stress tensor interacting with the mean (background) velocity gradient. This cross production component also inhibits the first mode. The dilatational pressure facilitates energy transfer from the kinetic to the internal field in the near‐wall region, whereas the energy transfer away from the wall is mostly due to the solenoidal pressure work. The dilatational characters of the fast and slow modes are also examined. The fast mode is dominantly dilatational at both $M=4$ and $M=6$, while the nature of the slow mode is dependent on $M$. Finally, Helmholtz decomposition of the perturbation momentum vector is performed. Interestingly, both first and second modes are dominated by solenoidal components of momentum.