Direct numerical simulations are conducted for temporally evolving stratified wake flows at Reynolds numbers from $10\,000$ to $50\,000$ and Froude numbers from $2$ to 50. Unlike previous studies that obtained statistics from a single realization, we take ensemble averages among 80–100 realizations. Our analysis shows that data from one realization incur large convergence errors. These errors reduce quickly as the number of statistical samples increases, with the benefit of ensemble average diminishing beyond 40–60 realizations. The data with ensemble average allow us to test the previously established scalings and arrive at new scaling estimates. Specifically, the data do not support power-law scaling in the centreline velocity deficit $U_0$ beyond the near wake. Its decay rate increases continuously from 0.1 at the onset of the non-equilibrium regime until the end of our calculations without reaching any asymptote. Additionally, while no power-law scalings could be found in the wake width ($L_H$) and wake height ($L_V$) in the late wake, $L_H\sim (Nt)^{1/3}$ is a good working approximation of the wake's horizontal size, where $N$ is buoyancy frequency and $t$ is time. Besides the low-order statistics, we also report the transverse integrated terms and the vertically integrated terms in the turbulent kinetic energy budget equation as a function of the vertical and transverse coordinates. The data indicate that there are two peaks in the vertically integrated production and transport terms, and one peak when the two terms are integrated horizontally.

Iceberg melting is a critical factor for climate change. However, the shape of an iceberg is an often neglected aspect of its melting process. Our study investigates the influence of different ice shapes and ambient flow velocities on melt rates by conducting direct numerical simulations of a simplified system of bluff body flow. Our study focuses on the ellipsoidal shape, with the aspect ratio as the control parameter. We found the shape plays a crucial role in the melting process, resulting in significant variations in the melt rate between different shapes. Without flow, the optimal shape for a minimal melt rate is the disk (two-dimensional) or sphere (three-dimensional), due to the minimal surface area. However, as the ambient flow velocity increases, the optimal shape changes with the aspect ratio. We find that ice with an elliptical shape (when the long axis is aligned with the flow direction) can melt up to 10 % slower than a circular shape when exposed to flowing water. Following the approach considered by Huang et al. (J. Fluid Mech., vol. 765, 2015, R3) for dissolving bodies, we provide a quantitative theoretical explanation for this optimal shape, based on the combined contributions from both surface-area effects and convective-heat-transfer effects. Our findings provide insight into the interplay between phase transitions and ambient flows, contributing to our understanding of the iceberg melting process and highlighting the need to consider the aspect-ratio effect in estimates of iceberg melt rates.

Recent microfluidic experiments have evidenced complex spatio-temporal fluctuations in low-Reynolds-number flows of polymer solutions through lattices of obstacles. However, understanding the nonlinear physics of such systems remains a challenge. Here, we use high performance simulations to study viscoelastic flows through a hexagonal lattice of cylindrical obstacles. We find that structures of localized polymer stress – in particular birefringent strands – control the stability and the dynamics. We first show that, at steady state, strands act as a web of sticky flow barriers that induce channelization, multistability and hysteresis. We then demonstrate that a spontaneous destabilization of the strands drives the transition to unsteady flow with regimes of self-sustained oscillations, travelling waves and strand pulsations. We further show that these pulsations, which result from the destabilization of envelope patterns of stress with strands wrapped around multiple obstacles, are integral to the transition towards elastic turbulence in our two-dimensional simulations. Our study provides a new perspective on the role of birefringent strands and a framework for understanding experimental observations. We anticipate that it is an important step towards unifying existing interpretations of the nonlinear physics of viscoelastic flows through complex structures.

The dynamics of soft porous media involves complex interactions between fluid flow and elasticity. The recent paper by Fiori et al. (J. Fluid Mech., vol. 974, 2023, A2) highlights phenomena relating to the periodic loading of such poro-elastic media, including hysteresis and the localisation of deformation at high frequencies. These effects could result in rectification and steady streaming in many important applications.

Experimental and theoretical studies on millimetre-sized droplets suggest that at low Reynolds number the difference between the drag force on a circulating water droplet and that on a rigid sphere is very small (less than 1 %) (LeClair et al., J. Atmos. Sci., vol. 29, 1972, pp. 728–740). While the drag force on a spherical liquid droplet at high viscosity ratios (of the liquid to the gas), is approximately the same as that on a rigid sphere of the same size, the other quantities of interest (e.g. the temperature) in the case of a rarefied gas flow over a liquid droplet differ from the same quantities in the case of a rarefied gas flow over a rigid sphere. The goal of this article is to study the effects of internal motion within a spherical microdroplet/nanodroplet – such that its diameter is comparable to the mean free path of the surrounding gas – on the drag force and its overall dynamics. To this end, the problem of a slow rarefied gas flowing over an incompressible liquid droplet is investigated analytically by considering the internal motion of the liquid inside the droplet and also by accounting for kinetic effects in the gas. Detailed results for different values of the Knudsen number, the ratio of the thermal conductivities and the ratio of viscosities are presented for the pressure and temperature profiles inside and outside the liquid droplet. The results for the drag force obtained in the present work are in good agreement with the theoretical and experimental results existing in the literature.

A uniform representation of the mean turbulent velocity profile in the sum of a wall function and a wake function, already introduced for other parallel geometries, is applied to an open channel. The open channel with its wake function is thus found to coherently fit in to the same theoretical picture previously drawn for plane Couette, plane closed-channel and circular-pipe flow, and to share with them a universal law of the wall and a universal logarithmic law with a common value of von Kármán's constant.

We present a suite of large-eddy simulations (LES) of a wind farm operating in conventionally neutral boundary layers. A fixed 1.6 GW wind farm is considered for 40 different atmospheric stratification conditions to investigate effects on wind-farm efficiency and blockage, as well as related gravity-wave excitation. A tuned Rayleigh damping layer and a wave-free fringe-region method are used to avoid spurious excitation of gravity waves, and a domain-size study is included to evaluate and minimize effects of artificial domain blockage. A fully neutral reference case is also considered, to distinguish between a case with hydrodynamic blockage only, and cases that include hydrostatic blockage induced by the air column above the boundary layer and the excitation of gravity waves therein. We discuss in detail the dependence of gravity-wave excitation, flow fields and wind-farm blockage on capping-inversion height, strength and free-atmosphere lapse rate. In all cases, an unfavourable pressure gradient is present in front of the farm, and a favourable pressure gradient in the farm, with hydrostatic contributions arising from gravity waves at least an order of magnitude larger than hydrodynamic effects. Using respectively non-local and wake efficiencies $\eta _{nl}$ and $\eta _{w}$, we observe a strong negative correlation between the unfavourable upstream pressure rise and $\eta _{nl}$, and a strong positive correlation between the favourable pressure drop in the farm and $\eta _{w}$. Using a simplified linear gravity-wave model, we formulate a simple scaling for the ratio $(1-\eta _{nl})/\eta _{w}$, which matches reasonably well with the LES results.

We employ a linear stability analysis and direct numerical simulations to study the characteristics of wall modes in thermal convection in a rectangular box under strong and inclined magnetic fields. The walls of the convection cell are electrically insulated. The stability analysis assumes periodicity in the spanwise direction perpendicular to the plane of a homogeneous magnetic field. Our study shows that for a fixed vertical magnetic field, the imposition of horizontal magnetic fields results in an increase of the critical Rayleigh number along with a decrease in the wavelength of the wall modes. The wall modes become tilted along the direction of the resulting magnetic fields and therefore extend further into the bulk as the horizontal magnetic field is increased. Once the modes localized on the opposite walls interact, the critical Rayleigh number decreases again and eventually drops below the value for onset with a purely vertical field. We find that for sufficiently strong horizontal magnetic fields, the steady wall modes occupy the entire bulk and therefore convection is no longer restricted to the sidewalls. The aforementioned results are confirmed by direct numerical simulations of the nonlinear evolution of magnetoconvection. The direct numerical simulation results also reveal that at least for large values of horizontal magnetic field, the wall-mode structures and the resulting heat transfer are dependent on the initial conditions.

An extended turbulent state can coexist with the stable laminar state in pipe flows. We focus here on short pipes with additional discrete symmetries imposed. In this case, the boundary between the coexisting basins of attraction, often called the edge of chaos, is the stable manifold of an edge state, which is a lower-branch travelling wave solution. We show that a low-dimensional submanifold of the edge of chaos can be constructed from velocity data using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of non-resonant spectral subspaces of the linearized system at stationary states. Using very low-dimensional SSM-based reduced-order models, we predict transitions to turbulence or laminarization for velocity fields near the edge of chaos.

Several methods have been proposed to characterize the complex interactions in turbulent wakes, especially for flows with strong cyclic dynamics. This paper introduces the concept of Fourier-averaged Navier–Stokes (FANS) equations as a framework to obtain direct insights into the dynamics of complex coherent wake interactions. The method simplifies the interpretations of flow physics by identifying terms contributing to momentum transport at different time scales. The method also allows for direct interpretation of nonlinear interactions of the terms in the Navier–Stokes equations. By analysing well-known cases, the characteristics of FANS are evaluated. Particularly, we focus on physical interpretation of the terms as they relate to the interactions between modes at different time scales. Through comparison with established physics and other methods, FANS is shown to provide insight into the transfer of momentum between modes by extracting information about the contributing pressure, convective and diffusive forces. The FANS equations provide a simply calculated and more directly interpretable set of equations to analyse flow physics by leveraging momentum conservation principles and Fourier analysis. By representing the velocity as a Fourier series in time, for example, the triadic model interactions are apparent from the governing equations. The method is shown to be applicable to flows with complex cyclic waveforms, including broadband spectral energy distributions.

In this paper, we present a theoretical, experimental and numerical study of the dynamics of cavitation bubbles inside a droplet suspended in another host fluid. On the theoretical side, we provided a modified Rayleigh collapse time and natural frequency for spherical bubbles in our particular context, characterized by the density ratio between the two liquids and the bubble-to-droplet size ratio. Regarding the experimental aspect, experiments were carried out for laser-induced cavitation bubbles inside oil-in-water (O/W) or water-in-oil (W/O) droplets. Two distinct fluid-mixing mechanisms were unveiled in the two systems, respectively. In the case of O/W droplets, a liquid jet emerges around the end of the bubble collapse phase, effectively penetrating the droplet interface. We offer a detailed analysis of the criteria governing jet penetration, involving the standoff parameter and impact velocity of the bubble jet on the droplet surface. Conversely, in the scenario involving W/O droplets, the bubble traverses the droplet interior, inducing global motion and eventually leading to droplet pinch-off when the local Weber number exceeds a critical value. This phenomenon is elucidated through the equilibrium between interfacial and kinetic energies. Lastly, our boundary integral model faithfully reproduces the essential physics of the non-spherical bubble dynamics observed in the experiments. We conduct a parametric study spanning a wide parameter space to investigate bubble–droplet interactions. The insights from this study could serve as a valuable reference for practical applications in the field of ultrasonic emulsification, pharmacy, etc.

Clustering of externally and evenly heated particles is enhanced by the increased viscosity of heated fluid in the vicinity of these clusters – a phenomenon known as viscous capturing (VC). Herein we study, via direct numerical simulations of decaying turbulence, the effect of temperature-driven viscosity on clustering with different particle loading densities. We employ a two-way momentum and energy coupling, and gas viscosity is modelled by a power law to understand the role of the increased drag and particle back-reaction force on the clustering intensity. For the continuum and dispersed phases, Eulerian and Lagrangian point particle schemes have been used, neglecting inter-particle collisions. We found that the enhanced viscosity-driven clustering is a strong function of particle loading density, as the increase in particle number density enables the formation of large uneven clusters before heating, which is the main condition for VC to take effect. Higher number density should result in greater turbulence modulation and negate local temperature-based viscous effects leading to VC. However, due to higher local particle number density in the clusters and interphase heat transfer, increased drag force prevails in such cases and delivers excessive clustering. By sampling conditionally the particle velocity and temperature inside the clusters, it is found that the thermodynamic and kinematic properties of the particles in the clusters are highly correlated, and this correlation increases with the particle loading density. Therefore, based on the particle number density, temperature-based viscosity can enhance considerably the clustering of heated particles and alter the effect of particles on the underlying turbulence.

We explore ${\rm CO}_2$ injection into a layered permeable rock consisting of high permeability reservoir layers, separated by low permeability mudstone, and taking the shape of an anticline within a laterally extensive aquifer. We first show how the storage capacity of the formation depends on the capillary entry pressure of the inter-layer mudstone, so that ${\rm CO}_2$ cannot flow from one layer into the next. We then consider a formation composed of two layers, overlain by a cap rock. For injection into the lowest layer, we show that the injection rate, capillary entry pressure and buoyancy driven flux through the mudstone determine whether the lower or upper layer fills to the spill point first. We also show that at the end of the injection phase, ${\rm CO}_2$ may continue to flow from the lower to the upper layer. This implies that injection should be stopped once the injected volume matches the static capacity of the formation in order to prevent spilling after injection. We present a series of analogue experiments of a two layered system that illustrate some of the principles described by the model, and assess the implications of the results for field scale systems.

In this paper a fully coupled particle-size segregation model for granular flows (Barker et al., J. Fluid Mech., vol. 909, 2021, p. A22) is used to simulate the development of the patterns in a triangular rotating drum. The results are compared with the experimental patterns formed with bidisperse and tridisperse granular mixtures, and with varying compositions and fill heights. In all cases the agreement between the simulations and experiments is remarkably good. The experimental patterns are generated in a narrow gap between transparent front and back sidewalls. These prevent three-dimensional motion, but also impose friction on the flow, making it thinner and faster than it would otherwise be. This promotes segregation, as it simultaneously increases the shear rate and reduces the local pressure. To obtain the correct flow dynamics and segregation, width-averaged sidewall friction is incorporated into the two-dimensional simulations, which are performed in OpenFOAM$^{\circledR}$. The free-surface avalanche forms a boundary layer within which all the segregation occurs. Material in the lower reach of the avalanche is continuously deposited into an underlying solid body of grains, which rotates with the drum, and is eventually re-entrained into the avalanche along its upper reach. The changing geometry of the granular region (as the drum rotates) implies that the avalanche is constantly adjusting its length, position and depth. This generates a complex quasi-periodic flow, which when combined with particle-size segregation generates amazing patterns in the solid rotating granular body after only two drum rotations.

Large eddy simulations (LES) are widely used to study the effects of surface morphology on turbulence statistics, exchange processes and turbulence topology in urban canopies. However, as LES are only approximations of reality, special attention is needed for the computational model set-up to ensure an accurate representation of the physical processes of interest. This paper shows that the choice of the numerical domain can significantly affect the accuracy of turbulent flow statistics, potentially causing a mismatch between numerical studies and experimental data. The study examines the influence of cross-stream aspect ratio (YAR), streamwise aspect ratio (XAR) and scale separation (SS) on first- and second-order flow statistics and turbulence topology. It is found that domains with a low YAR underestimate the velocity variance, while those with a low XAR overestimate the variance value. The study proposes a new approach based on the Buckingham Pi theorem to evaluate the effect of SS, as the existing method has major limitations for canopy flows. The results suggest that domains with small SS underpredict the variance value. To minimise the artificial impact of the numerical domain on turbulent flow statistics, the study recommends guidelines for future research, including a YAR of 3 or more, an XAR of 6 or more and an SS of 12 or more. Error tables are presented to allow researchers to select smaller domains than recommended, depending on their research interests in specific parts of the flow.

As presented in Annenkov & Shrira (Phys. Rev. Lett., vol. 102, 2009, 024502), when a surface gravity wave field is subjected to an abrupt perturbation of external forcing, its spectrum evolves on a ‘fast’ dynamic time scale of $O(\varepsilon ^{-2})$, with $\varepsilon$ a measure of wave steepness. This observation poses a challenge to wave turbulence theory that predicts an evolution with a kinetic time scale of $O(\varepsilon ^{-4})$. We revisit this unresolved problem by studying the same situation in the context of a one-dimensional Majda–McLaughlin–Tabak equation with gravity wave dispersion relation. Our results show that the kinetic and dynamic time scales can both be realised, with the former and latter occurring for weaker and stronger forcing perturbations, respectively. The transition between the two regimes corresponds to a critical forcing perturbation, with which the spectral evolution time scale drops to the same order as the linear wave period (of some representative mode). Such fast spectral evolution is mainly induced by a far-from-stationary state after a sufficiently strong forcing perturbation is applied. We further develop a set-based interaction analysis to show that the inertial-range modal evolution in the studied cases is dominated by their (mostly non-local) interactions with the low-wavenumber ‘condensate’ induced by the forcing perturbation. The results obtained in this work should be considered to provide significant insight into the original gravity wave problem.

There are numerous examples of long-lived, surface-intensified anticyclones over submarine depressions and troughs in the ocean. These often co-exist with a large-scale cyclonic circulation. The latter is predicted by existing barotropic theory but the anticyclone is not. We extend one such theory, which minimizes enstrophy while conserving energy, to two fluid layers. This yields a bottom-intensified flow with cyclonic circulation over a depression. The solution is steady, an enstrophy minimum and stable. When the Lagrange multiplier, $\lambda$, is near zero, the total potential vorticity (PV) becomes homogenized, in both layers. For positive $\lambda$, the surface PV is anticyclonic and strongest at intermediate energies. In quasi-geostrophic numerical simulations with a random initial perturbation PV, the bottom-intensified cyclonic flow always emerges. Vortices evolve independently in the layers and vortex mergers are asymmetric over the depression; cyclones are preferentially strained out at depth while only anticyclones merge at the surface. Both asymmetries are linked to the topographic flow. The deep cyclones feed the bottom-intensified cyclonic circulation while the asymmetry at the surface is only apparent after that circulation has spun up. The result of the surface merger asymmetry is often a lone anticyclone above the depression. This occurs primarily at intermediate energies, when the surface PV predicted by the theory is strongest. Similar results obtain in a full complexity ocean model but with a more pronounced asymmetry in surface vortex mergers and, with bottom friction, significant bottom flow beneath the central anticyclone.

The near wake of a small-scale wind turbine is investigated using particle image velocimetry experiments at different tip speed ratios ($\lambda$). The wind turbine model had a nacelle and a tower mimicking real-scale wind turbines. The near wake is found to be dominated by multiple coherent structures, including the tip vortices, distinct vortex sheddings from the nacelle and tower, and wake meandering. The merging of the tip vortices is found to be strongly dependent on $\lambda$. A convective length scale ($L_c$) related to the pitch of the tip vortices is defined that is shown to be a better length scale than turbine diameter ($D$) to demarcate the near wake from the far wake. The tower induced strong vertical asymmetry in the flow by destabilising the tip vortices and promoting mixing in the lower (below the nacelle) plane. The nacelle's shedding is found to be important in ‘seeding’ wake meandering, which, although not potent, exists close to the nacelle, and it becomes important only after a certain distance downstream ($x>3L_c$). A link between the ‘effective porosity’ of the turbine and $\lambda$ is established, and the strength and frequency of wake meandering are found to be dependent on $\lambda$. In fact, a decreasing trend of wake meandering frequency with $\lambda$ is observed, similar to vortex shedding from a porous plate at varying porosity. Such similarity upholds the notion of wake meandering being a global instability of the turbine, which can be considered as a ‘porous’ bluff body of diameter $D$.

We revisit the problem of a solid sphere rising slowly in a rotating short container filled with a slightly viscous fluid, with emphasis on the drag force. The data of the classical experiments of Maxworthy (J. Fluid Mech., vol. 31, 1968, pp. 643–655) and recent experiments of Kozlov et al. (Fluids, vol. 8 (2), 2023, paper 49), and the available geostrophic and quasi-geostrophic theories, are subjected to a novel scrutiny by combined reprocessing and comparisons. The measured drag is, consistently, about 20 % lower than the geostrophic prediction (assuming that flow is dominated by the Ekman layers, while in the inviscid cores the Coriolis acceleration is supported by the pressure gradient). The major objective is the interpretation and improvement of the gap between data and predictions. We show that the data cover a small range of relevant parameters (in particular the Taylor number $T$ and the height ratio $H$ of cylinder to particle diameter) that precludes a thorough and reliable assessment of the theories. However, some useful insights and improvements can be derived. The hypothesis that the discrepancy between data and the geostrophic prediction is due to inertial effects (not sufficiently small Rossby number $Ro$ in the experiments) is dismissed. We show that the major reason for the discrepancy is the presence of relatively thick Stewartson layers about the cylinder (Taylor column) attached to the sphere. The $1/3$ layer displaces the boundary condition of the angular velocity ($\omega = 0$) outside the radius of the particle. This observation suggests a semi-empirical correction to the theoretical quasi-geostrophic predictions (which takes into account the Ekman layers and the $1/4$ Stewartson layers); the corrected drag is in fair agreement with the data. We demonstrate that the inertial terms are negligible for $Ro\,T^{1/2} <0.4$. We consider curve-fit approximations, and point out some persistent gaps of knowledge that require further experiments and simulations.