Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

Eigenvectors of the observability and controllability Gramians represent responsive and receptive flow structures that enjoy a well-established connection to resolvent forcing and response modes. However, whereas resolvent modes have demonstrated great potential to guide sensor and actuator placement, observability and controllability modes have been leveraged exclusively in the context of model reduction via input and output projections. In this work, we introduce interpolatory, rather than orthogonal, input and output projections, that can be leveraged for sensor and actuator placement and open-loop control design. An interpolatory projector is an oblique projector with the property of preserving certain entries in the vector being projected. We review the connection between the resolvent operator and the Gramians, and present several numerical examples where we perform both orthogonal and interpolatory input and output projections onto the dominant forcing and response subspaces. Input projections are used to identify dynamically relevant disturbances, place sensors to measure disturbances, and place actuators for feedforward control in the linearized Ginzburg–Landau equation. Output projections are used to identify coherent structures and place sensors aiming at state reconstruction in the turbulent flow in a minimal channel at $Re_{\tau }=185$. The framework does not require data snapshots and relies only on knowledge of the steady or mean flow.

A novel experiment was performed in rotating Rayleigh–Bénard convection (RRBC), wherein the convection cell with radius $R$ was shifted away from the rotation axis by a distance $d$. In this case, the centrifugal force felt by a fluid parcel (characterized by the Froude number $Fr$) can be decomposed into an axisymmetrical part $Fr_R$ and a directed one $Fr_d$. It has been reported that the global heat transport enhances at $Fr_{d,c}$ and then reaches an optimal state at $Fr_{d,max}$ (Hu et al., Phys. Rev. Lett., vol. 127, 2021, 244501). In this paper, the local properties after the offset effects set in are investigated further, which show different features before and after $Fr_{d,max}$. The local temperature measurements at the cell centre reveal that the bulk flow turns from a turbulent state into a laminar state at $Fr_{d,max}$, which is consistent with the particle image velocimetry results. This transition can be qualitatively understood by an equivalent tilted RRBC system. As for the hot and cold coherent structures near the sidewall, their vertical temperature variations reach a minimum at $Fr_{d,max}$, implying that these structures are mostly uniform in the vertical direction at $Fr_{d,max}$. Their temperature contrasts show a linear dependence on $Fr_d$ and start to deviate from this linear behaviour when $Fr_d>Fr_{d,max}$. Besides the dominant effects of $Fr_d$, the secondary effects of $Fr_R$ are also investigated. Due to the positive effect of $+Fr_R$ on the cold structure and the negative effect of $-Fr_R$ on the hot one, the cold structure is more coherent than the hot one, but its size is smaller. The shift of the cold cluster centre from the farthest point is also larger than the shift of the hot one from the nearest point.

Using laboratory experiments and numerical simulations, we examine the transfer of soluble material from small, spherical particles sinking in homogeneous turbulence at large Péclet number. A theoretical analysis predicts two distinct mechanisms of convective mass transfer: strain due to turbulence and slip due to gravitational settling. Their relative strength is parametrised by the sinking ratio, ${Sr} = w_0 \tau _\eta /a$, where $w_0$ is the quiescent settling velocity, $a$ is the particle radius and $\tau _\eta$ is the Kolmogorov time scale. This analysis predicts that the topology of the concentration wake changes from a symmetric topology at ${Sr} \ll 1$ to an asymmetric topology at ${Sr} \gg 1$ as the dominant mechanism of mass transfer changes. Particle tracking flow visualisations of small spheres releasing dye in turbulence confirm the existence of this change in mechanism at ${Sr} = O(1)$. We complement these experiments with numerical simulations of the mass transfer from sinking particles. The transfer rate predicted by the simulation is found to be in good agreement with literature data for mass transfer to turbulent suspensions of solid particles and is consistent with asymptotic expressions for mass transfer in uniform flow when ${Sr} \gg 1$. A decomposition of the convective fluxes confirms the transition in the transfer mechanism. At ${Sr} = O(1)$, both mechanisms provide comparable contributions to the transfer rate. Cross-correlation analysis reveals that particle-scale knowledge of both the recent strain and velocity history is required to predict the instantaneous transfer rate. Turbulence-induced particle rotation has a modest suppression effect upon convective transfer by sinking.

Magneto-gravity-elliptic instability is addressed here considering an unbounded strained vortex (with constant vorticity $2\varOmega$ and with ellipticity parameter $\varepsilon$) of a perfectly conducting fluid subjected to a uniform axial magnetic field (with Alfvén velocity scaled from the basic magnetic field $B)$ and an axial stratification (with a constant Brunt–Väisälä frequency $N$). Such a simple model allows us to formulate the stability problem as a system of equations for disturbances in terms of Lagrangian Fourier (or Kelvin) modes which is universal for wavelengths of the perturbation sufficiently small with respect to the scale of variation of the basic velocity gradients. It can model localised patches of elliptic streamlines which often appear in some astrophysical flows (stars, planets and accretion discs) that are tidally deformed through gravitational interaction with other bodies. In the limit case where the flow streamlines are exactly circular ($\varepsilon =0),$ there are fast and slow magneto-inertia-gravity waves with frequencies $\omega _{1,2}$ and $\omega _{3,4},$ respectively. Under the effect of finite ellipticity, the resonant cases of these waves, $\omega _i-\omega _j=n\varOmega$ $(i\ne j)$ ($n$ being an integer), can become destabilising. The maximal growth rate of the subharmonic instability (related to the resonance of order $n=2)$ is determined by extending the asymptotic method by Lebovitz & Zweibel (Astrophys. J., vol. 609, 2004, pp. 301–312). The domains of the $(k_0B/\varOmega, N/\varOmega )$ plane for which this instability operates are identified ($1/k_0$ being a characteristic length scale). We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). The axial stable stratification enhances the subharmonic instability related to the resonance between two slow modes because, at large magnetic field strengths, its maximal growth rate is twice that found in the case without stratification.

Time-resolved particle image velocimetry measurements were performed downstream of a swept backward-facing step. The measurements allow detailed analysis of the interactions between the unsteady instabilities and the stationary crossflow vortices. Different mechanisms are identified that lead to the modulation of the different families of unsteady instabilities that occur downstream of the step. For the low-frequency spanwise-travelling mode, the modulation occurs due to a redistribution of momentum when the instability encounters regions of large spanwise shear of the wall-normal and streamwise velocity. However, the higher-frequency streamwise-travelling instabilities undergo the familiar ‘lift-up’ mechanism when they encounter the regions of large vertical velocity due to the presence of the stationary crossflow vortices. The process leading to large velocity spikes, and ultimately to a laminar breakdown to turbulence, is identified as a constructive interaction between the different unsteady instabilities, coupled with an interaction with the stationary crossflow vortices when the phases align properly.

We present results of numerical simulations of a stratified reservoir with a three-layer stratification, subject to an oscillating surface shear stress. We investigate the effect of sloped endwalls on mixing and internal wave adjustment to forcing within the basin, for three different periods of forcing. The simulations are carried out at a laboratory scale, using large-eddy simulation. We solve the three-dimensional Navier–Stokes equations under the Boussinesq approximation using a second-order-accurate finite-volume solver. The model was validated by reproducing experimental results for the response of a reservoir to surface shear stress and resonant frequencies of internal waves. We find interesting combinations of wave modes and mixing under variation of the forcing frequencies and of the inclination of the endwalls. When the frequency of the forcing is close to the fundamental mode-one wave frequency, a resonant internal seiche occurs and the response is characterized by the first vertical mode. For forcing periods twice and three times the fundamental period, the dominant response is in terms of the second vertical mode. Adjustment to forcing via the second vertical mode is accompanied by the cancellation of the fundamental wave and energy transfer to higher-frequency waves. The study shows that the slope of the endwalls dramatically affects the location of mixing, which has a feedback on the wave field by promoting the generation of higher vertical modes.

This work deals with the linear surface waves generated by a vessel advancing at a constant forward speed. These waves, known as ship waves, appear stationary to an observer on the vessel. Rather than exploring the well-studied stationary ship waves, this work delves into the physical properties of ship waves measured at Earth-fixed locations. While it might have been expected that analysing these waves in an Earth-fixed coordinate system would be a straightforward transformation from existing analytical theories in a moving coordinate system, the reality proves to be quite different. The properties of waves measured at fixed locations due to a passing ship turn out to be complex and non-trivial. They exhibit unique characteristics, being notably unsteady and short crested, despite appearing stationary to an observer on the generating vessel. The analytical expressions for the physical properties of these unsteady waves are made available in this work, including the amplitude, frequency, wavenumber, direction of propagation, phase velocity and group velocity. Based on these newly derived expressions and two-point measurements, an inverse method has been presented for determining the advancing speed and the course of motion of the moving ship responsible for the wave generation. The results from this study can be used in a wide range of applications, such as interpreting data from point measurements and assessing the roles of ship waves in transporting ocean particles.

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$, compares to the basic-state amplitude parameter, $\epsilon \ll 1$. For $\mu ={O}(1)$, the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$, this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

The fluid–structure interaction problem composed of an elongated, finite-length, flexible cylinder falling in a fluid at rest is investigated experimentally. Tomographic reconstruction of the cylinder and three-dimensional particle tracking velocimetry of the surrounding fluid, based on the Shake-The-Box algorithm, are used jointly to capture both solid and fluid motions. Starting from the rectilinear vertical fall characterized by a steady wake, focus is placed on subsequent regimes involving, mainly in the horizontal direction, periodic rigid-body motions (RBM) of weak amplitude or periodic large-amplitude bending oscillations (BO). Two RBM regimes are explored: the TRA regime where the cylinder exhibits translational oscillations in a plane perpendicular to its axis, and the AZI regime in which the body displays an azimuthal oscillation around its centre. The associated unsteady wakes are composed of counter-rotating vortices bending near the body ends to connect with the adjacent vortex rows. Specific organizations of the vortical structures are uncovered, depending on the regime. In particular, in the AZI regime, they present an antisymmetrical distribution relative to the midspan point. For a sufficiently long cylinder, BO regimes emerge, resembling the structural modes of an unsupported beam. The associated wakes exhibit a cellular organization. Within each cell delimited by two deformation nodes, two counter-rotating vortex rows are shed per oscillation cycle. Flow velocity fluctuations are in phase opposition on each side of a deformation node. For both RBM and BO regimes, frequency and phase analyses of cylinder and wake behaviours, along the span, highlight the spatio-temporal synchronization of the unsteady flow and moving body.

Coherent structures in two-dimensional Navier–Stokes turbulence are ubiquitously observed in nature, experiments and numerical simulations. The present study conducts a comparison between several structure detection schemes based on the Okubo–Weiss criterion, the vorticity magnitude and Lagrangian coherent structures (LCSs), focusing on the inverse cascade in two-dimensional hydrodynamic turbulence. A recently introduced vortex scaling phenomenology (Burgess & Scott, J. Fluid Mech., vol. 811, 2017, pp. 742–756) allows the quantification of the respective thresholds required by these methods based on physical properties of the flow. The resulting improved comparability allows us to identify characteristic relative differences in the detection sensitivity between the employed structure detection techniques. With respect to the inverse cascade of energy, coherent structures contribute, as expected, substantially less to the cross-scale flux than the residual incoherent parts of the flow although the energetically dominant coherent structures lead to an important large-scale deformation of the energy spectrum. This cascade inactivity can be understood by an increased misalignment of strain-rate and subgrid stress tensors within coherent structures. At the same time, the structures exhibit strong and localised nonlinear cross-scale interactions that appear to stabilise them. We quantify and interpret the resulting shape preservation of coherent structures in terms of a multi-scale gradient approach (Eyink, J. Fluid Mech., vol. 549, 2006, pp. 191–214) as the depletion of strain rotation and vorticity gradient stretching whereas the dynamics of the residual fluctuations are consistent with the vortex thinning picture.

The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier–Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian. A recent model handles these unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson & Meneveau, Phys. Rev. Fluids, vol. 2 (7), 2017, 072601). This model is in remarkable agreement with direct numerical simulations (DNS) data and works for arbitrary Taylor Reynolds numbers $\textit {Re}_\lambda$. Inspired by this, we develop a Lagrangian model for passive scalar gradients in isotropic turbulence. The equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term, which we model using the RDGF approach. However, comparisons of the statistics obtained from this model with DNS data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to $\textit {Re}_\lambda \approx 500$, beyond this, the model breaks down.

Modelling the flow of dry granular materials entering water is crucial for optimizing blending processes in industry and for natural hazard assessment when describing tsunami waves induced by landslides. In this study, we experimentally investigate the case of a jet of grains entering from the air into a water bath. After an initial transient state, a stationary impregnation front appears between the dry and the wet grains. The wet grains are then dispersed in the liquid. To describe this dry-to-wet transition, we focus on the first step of the process, when the liquid invades the dense granular medium. In this regime, the granular jet is modelled as a translating porous material, and we systematically characterize the impregnation process using a combination of experiments, analytical modelling, and numerical tools. We then compare this process to the situation of a confined granular jet entering a water bath. Our approach is a first step toward describing the interplay between dry grains and a liquid and the resulting dispersion of particles.

We consider inertial waves propagating in a fluid contained in a non-axisymmetric three-dimensional rotating cavity. We focus on the particular case of a fluid enclosed inside a truncated cone or frustum, which is the volume that lies between two horizontal parallel planes cutting an upright cone. While this geometry has been studied in the past, we generalise it by breaking its axisymmetry and consider the case of a truncated elliptic cone for which the horizontal sections are elliptic instead of circular. The problem is first tackled using ray tracing, where local wave packets are geometrically propagated and reflected within the closed volume without attenuation. We complement these results with a local asymptotic analysis and numerical simulations of the original linear viscous problem. We show that the attractors, well known in two dimensional or axisymmetric domains, can be trapped in a particular plane in three dimensions provided that the axisymmetry of the domain is broken. Contrary to previous examples of attractors in three-dimensional domains, all rays converge towards the same limit cycle regardless of initial conditions, and it is localised in the bulk of the fluid.

A surfactant-laden liquid film that lines the inside of an oscillating spherical cap is considered as a model of lung alveoli. Pulmonary surfactant solubility is described by Langmuir adsorption kinetics, modified by incorporating the intrinsic compressibility of the adsorbed monolayer. A novel boundary condition, supported by experimental data and scaling arguments, is applied at the rim. The condition enforces mass conservation of water and surfactant by matching the ‘large-scale’ dynamics of the alveolus to ‘small-scale’ equilibrium over mid-alveolar septa of small but finite thickness. Linear and weakly nonlinear analysis around the conditions in a non-oscillating cap indicates that the occurrence of shearing motion in the liquid is related to the non-zero film thickness over the rim, and shearing velocity at the interface is predicted an order-of-magnitude lower than the velocity of radial oscillation. Marangoni stresses dominate the interfacial dynamics, but capillary stresses affect significantly the interior flow field. In particular, they produce spatial modulations in flow rate, surface concentration of surfactant and wall shear stress, whose length scale varies with $Ca^{-1/3}$, i.e. is determined by a balance between capillary and viscous forces. Non-zero adsorption kinetics modifies at first order only the amplitude and phase of surface concentration, but affects all other variables at second order. In particular, it sets a steady drift of surfactant away from the alveolus and towards the rim. Finally, an attempt is made to relate the present predictions to physiological findings about air flow and particle deposition inside alveoli, and about shear stress-inflicted damage in diseased lungs.

Just as a solid object would, a liquid jet or a stream of droplets impacting a free surface deforms and perforates it. This generic flow interaction, met in everyday life but also in cutting edge industrial processes, has puzzled scientists for centuries. Lee et al. (J. Fluid Mech., vol. 921, 2021, A8) present an experimental study of a simple droplet train interacting with a liquid bath and identify two stages in the interaction: a first where a cavity elongates and finally bursts, and a second where the interface is steadily punched by the incoming stream. Each of these regimes is explained with elementary but effective models arising from first principles, thereby revealing a full and simple picture of the physics of making holes in liquids.

Vertical convection is the fluid motion that is induced by the heating and cooling of two opposed vertical boundaries of a rectangular cavity (see e.g. Wang et al., J. Fluid Mech., vol. 917, 2021, A6). We consider the linear stability of the steady two-dimensional flow reached at Rayleigh numbers of O($10^8$). As a function of the Prandtl number, $Pr$, and the height-to-width aspect ratio of the domain, $A$, the base flow of each case is computed numerically and linear simulations are used to obtain the properties of the leading linear instability mode. Flow regimes depend on the presence of a circulation in the entire cavity, detachment of the thermal layer from the boundary or the corner regions and on the oscillation frequency relative to the natural frequency of oscillation in the stably temperature-stratified interior, allowing for the presence of internal waves or not. Accordingly, the regime is called slow or fast, respectively. Either the global circulation or internal waves in the interior may couple the top and bottom buoyancy currents, while their absence implies asymmetry in their perturbation amplitude. Six flow regimes are found in the range of $0.1 \leq Pr \leq 4$ and $0.5 \leq A \leq 2$. For $Pr \lessapprox 0.4$ and $A>1$, the base flow is driven by a large circulation in the entire cavity. For $Pr \gtrapprox 0.7$, the thermal boundary layers are thin and the instability is driven by the motion along the wall and the detached boundary layer. A transition between these regimes is marked by a dramatic change in oscillation frequency at $Pr = 0.55 \pm 0.15$ and $A <2$.

In nature and engineering applications, water jet plunging acts as a key process causing interface breaking and generating mixed-phase turbulence. In this paper, high-resolution numerical simulations of the plunging of a water jet into a quiescent pool were performed to investigate the statistical properties of mixed-phase turbulence, with a special focus on the closure problem of the Reynolds-averaged equation. We conducted phase-resolved simulations, with the air–water interface captured using a coupled level-set and volume-of-fluid method. Various cases were performed to analyse the effects of the Froude number and Reynolds number. The simulation results showed that the turbulence statistics are insensitive to the Reynolds number under investigation, while the Froude number influences the flow properties significantly. To investigate the closure problem of the mean momentum equation, the turbulent kinetic energy (TKE) and turbulent mass flux (TMF) and their transport equations were analysed further. It was discovered that the balance relationship of the TKE budget terms remained similar to many single-phase turbulent flows. The TMF is an additional unclosed term in mixed-phase turbulence over the single-phase turbulence. Our simulation results showed that the production term in its transport equation was highly correlated to TKE. Based on this finding, a closure model for the production term of TMF was further proposed.

We investigate the equilibrium of an axisymmetric system consisting of sessile and pendent drops on pre-stretched nonlinear elastic membranes. The membrane experiences large deformations due to the drop's weight and interfacial interactions. We first show that force balance alone leads to non-unique equilibrium solutions. Identifying the system's equilibrium with the minimum of its free energy, we then demonstrate that the equilibrium solution is made unique by requiring the continuity of meridional stretches across the three-phase contact circle. For a special class of nonlinear elastic materials – $I_2$ materials – we then compute the equilibrium configurations of the drop–membrane system for a range of drop volumes and membrane pre-tensions. Finally, the present work facilitates two important applications: (a) the membrane's pre-tension and current tension are related exactly to help in utilizing the system as an elastocapillary probe for membrane pre-tension and (b) we suggest an experimental protocol for measuring the membrane's surface properties.