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.

The highly nonlinear evolution of the single-mode stratified compressible Rayleigh–Taylor instability (RTI) is investigated via direct numerical simulation over a range of Atwood numbers ($A_T=0.1$–$0.9$) and Mach numbers ($Ma=0.1$–$0.7$) for characterising the isothermal background stratification. After the potential stage, it is found that the bubble is accelerated to a velocity which is well above the saturation value predicted in the potential flow model. Unlike the bubble re-acceleration behaviour in quasi-incompressible RTI with uniform background density, the characteristics in the stratified compressible RTI are driven by not only vorticity accumulation inside the bubble but also flow compressibility resulting from the stratification. Specifically, in the case of strong stratification and high $A_T$, the flow compressibility dominates the bubble re-acceleration characters. To model the effect of flow compressibility, we propose a novel model to reliably describe the bubble re-acceleration behaviours in the stratified compressible RTI, via introducing the dilatation into the classical model that takes into account only vorticity accumulation.

Classical theories for suspensions have been formulated by starting from the Navier–Stokes equations describing pure liquid flow and then introducing additional dependencies to account for the presence of suspended particles. These models are often accurate for low particle concentrations but have lacked a convincing description of the frictional interactions of particles, which are important at larger solid volume fractions. The $\mu (J), \varPhi (J)$ rheology, which draws a direct analogy between suspension flow and dry granular flow, is a recent theory that addresses this issue, but is shown here to be dynamically ill-posed for large solid volume fractions. An alternative well-posed theory is introduced that includes additional dependence on the particle-phase dilation and compression. The new theory, denoted vCIDR, is tested numerically to show grid convergence for problems in which the $\mu (J), \varPhi (J)$ rheology instead suffers from catastrophic blow-up. A further well-posed extension provides a framework for handling the transition between viscous and inertial flows.

We study how channel width variations influence the dynamics of free-surface granular flows. For this purpose, we extend a continuum model framework to granular flows passing through channels that narrow or widen. Our theory uses a linearized approximation to an established dense granular flow rheology and a Coulomb friction law to model interaction between flow and sidewalls. We test the theoretical predictions using two novel 40 cm-diameter drums (convex and concave) filled halfway with 2 mm diameter particles rotated at rates in which the shear layer remains shallow and dense. We apply particle tracking velocimetry to enable quantitative comparisons between experimental data and theoretical predictions. We find that our experimental kinematics and energy profiles largely agree with the theoretical predictions. In general, flows through narrowing channels are faster and deeper than flows through widening channels. The influence of width variations grows with increasing flow speed, and the form of the rate dependence changes fundamentally as the regime changes from one in which kinetic energy is dissipated locally to one in which it is advected downstream. For both regimes, theoretical scaling analysis leads us to experimentally validated power laws, in which the exponent depends on the flow regime, and the multiplicative coefficient depends on channel geometry alone. Finally, we discuss how the differences between theoretical predictions and experimental data may be useful for improving our understanding of flows through non-uniform channels.

Experiments are conducted in water films falling along the bottom wall of a weakly inclined rectangular channel of height $5$ mm in the presence of a laminar counter-current air flow. Boundary conditions have been specifically designed to avoid flooding at the liquid outlet, thus allowing us to focus on the wave dynamics in the core of the channel. Surface waves are excited via coherent inlet forcing before they come into contact with the air flow. The effect of the air flow on the height, shape and speed of two-dimensional travelling nonlinear waves is investigated and contrasted with experiments of Kofman, Mergui & Ruyer-Quil (Intl J. Multiphase Flow, vol. 95, 2017, pp. 22–34), which were performed in a weakly confined channel. We observe a striking difference between these two cases. In our strongly confined configuration, a monotonic stabilizing effect or a non-monotonic trend (i.e. the wave height first increases and then diminishes upon increasing the gas flow rate) is observed, in contrast to the weakly confined configuration where the gas flow is always destabilizing. This stabilizing effect implies the possibility of attenuating waves via the gas flow and it confirms recent numerical results obtained by Lavalle et al. (J. Fluid Mech., vol. 919, 2021, R2) in a superconfined channel.

We consider closed-loop control of a two-dimensional supersonic boundary layer at $M=4.5$ that aims at reducing the linear growth of second Mack mode instabilities. These instabilities are first characterized with local spatial and global resolvent analyses, which allow us to refine the control strategy and to select appropriate actuators and sensors. After linear input–output reduced-order models have been identified, multi-criteria structured mixed $H_{2}$/$H_{\infty }$ synthesis allows us to fix beforehand the controller structure and to minimize appropriate norms of various transfer functions: the $H_{2}$ norm to guarantee performance (reduction of perturbation amplification in nominal condition), and the $H_{\infty }$ norm to maintain performance robustness (with respect to sensor noise) and stability robustness (with respect to uncertain free-stream velocity/density variations). Both feedforward and feedback set-ups, i.e. with estimation sensor placed respectively upstream/downstream of the actuator, allow us to maintain the local perturbation energy below a given threshold over a significant distance downstream of the actuator, even in the case of noisy estimation sensors or free-stream density variations. However, the feedforward set-up becomes completely ineffective when convective time delays are altered by free-stream velocity variations of $\pm$5 %, which highlights the strong relevance of the feedback set-up for performance robustness in convectively unstable flows.

We consider the propagation of inviscid bubbles in a Hele-Shaw cell under a uniform background flow. We focus on the distinguished limit in which the hydrodynamic pressure gradient due to the external flow balances viscous drag effects due to thin liquid films between the bubbles and the cell walls (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188), with the ratio between these two effects measured by a single dimensionless parameter that we label $\delta$. In this regime, we find that each bubble remains approximately circular, and its propagation velocity is determined by a net force balance. The analytical solution for the problem of an isolated bubble in an infinite Hele-Shaw cell is found to agree well with experimental data in the literature. In particular, we find that the bubble may move faster or slower than the background fluid speed, depending on whether $\delta >1$ or $\delta <1$, or precisely with the background flow if $\delta =1$. When the model is generalised to include the effects of multiple bubbles and boundaries in the Hele-Shaw cell, we still find that the sign of $\delta -1$ causes striking changes in the qualitative behaviour. For a train of three or more bubbles moving along a Hele-Shaw channel, we observe longitudinal waves that propagate forwards or backwards along the bubble train, depending on whether $\delta >1$ or $\delta <1$, resembling a Hele-Shaw Newton's cradle.

A two-dimensional system with a floating cylinder confined between two parallel vertical stationary plates partially immersed in an infinite liquid bath in a downward gravity field is considered. The equilibrium states of the system are investigated using the Young–Laplace equation in two dimensions. According to the symmetry of menisci at both sides of the cylinder, the equilibrium states are classified into three types: the equilibria of fully symmetric menisci, the equilibria of partially symmetric menisci and the equilibria of asymmetric menisci. The study is then extended to investigate the stabilities of the confined cylinder with bifurcation theory. Results show that there can be at most two stable regions in the bifurcation diagram. For different plates’ contact angles, there are five representative types of bifurcation behaviours for either the case of one stable region or the case of two stable regions. In comparison with the case of an unconfined cylinder, the confinement by two hydrophobic plates with a small spacing can assist the stable interfacial floatation of the confined cylinder with a large weight.

A framework for the budget of Reynolds stress, temperature and density variations, turbulent mass flux and turbulent heat flux is presented for buoyancy-generated turbulent flows with a focus on turbulent plumes. The dynamical interactions between the turbulence generated from the velocity and thermodynamic fluctuations in thermal and heated gas plumes is studied. A large-eddy simulation tool has been used to simulate heavier and lighter than air plumes released from a circular heated source for high Reynolds number (Re). The budget equations are developed using a Favre-averaging approach. Both heated air plumes and heated gas plumes, namely, heated ${\rm SO}_2$ and heated ${\rm CH}_4$ plumes, are included in the study. The study focuses on addressing key questions – does the turbulence kinetic energy (TKE) spectrum follow the classical inertial $-5/3$ spectrum or is there a $-3$ buoyancy regime with a different scaling law? Fundamental pathways for generation of turbulence through velocity fluctuations, and temperature and density fluctuations are discussed. Analysis of the TKE and scalar variance budget equations has demonstrated that multiple mechanisms dominate the transport of Reynolds stresses, and in particular that correlations between the density and the velocity dominates the turbulence production mechanisms. Entrainment is connected to the turbulent transport and pressure-dilation processes.

A two-dimensional depth-averaged model is derived for open-channel flows in the smooth turbulent case. The derivation is consistently obtained with a method of matched asymptotic expansions in the outer and inner layers using a mixing length model of turbulence including the free-surface reduction of the eddy viscosity. The shear effects are taken into account by an extra tensor variable of the model called the enstrophy tensor. The friction coefficient is an explicit expression of the water depth. The three-dimensional velocity field and the friction velocity can be reconstructed from the values of the depth-averaged quantities. Numerical simulations show that the enstrophy can be used to evaluate the development of the turbulent boundary layer. In the case of subcritical unsteady flows, the reconstructed velocity can be described with a logarithmic law modified by Coles’ wake function with apparent von Kármán constant, integration constant and wake-strength parameter, which differ from their values in steady flows. In the viscous sublayer the steady-state relation between the velocity and the vertical coordinate, in the inner scaling, is not valid for unsteady flows. Large errors on the calculation of the von Kármán constant can be made if the validity of the steady-state relation is assumed for unsteady flows. The comparisons of the reconstructed velocity profiles in the case of one-dimensional unsteady open-channel flows and two-dimensional wide trapezoidal channels show a good agreement with experiments.

We compute the full set of second-order inertial corrections to the instantaneous force and torque acting on a small spherical rigid particle moving unsteadily in a general steady linear flow. This is achieved by using matched asymptotic expansions and formulating the problem in a coordinate system co-moving with the background flow. Effects of unsteadiness and fluid-velocity gradients are assumed to be small, but to dominate in the far field over those of the velocity difference between the body and fluid, making the results essentially relevant to weakly positively or negatively buoyant particles. The outer solution (which at first order is responsible for the Basset–Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or angular acceleration between the particle and fluid, while others take the form of products of the rotation/strain rate of the background flow and the relative translational or angular velocity between the particle and fluid. Adding the outer and inner contributions, the known added-mass force or the spin-induced lift force are recovered, and new effects involving the velocity gradients of the background flow are revealed. The resulting force and torque equations provide a rational extension of the classical Basset–Boussinesq–Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow.

Flow transitions are an important fluid-dynamic phenomena for many reasons, including the direct effect on the aerodynamic forces acting on the body. In the present study, two-dimensional (2-D) and three-dimensional (3-D) wake transitions of a NACA0012 airfoil are studied for angles of attack in the range $0^\circ \leq \alpha \leq 20^\circ$ and Reynolds numbers $500 \leq {\textit {Re}} \leq 5000$. The study uses water-channel experiments and 2-D and 3-D numerical simulations based on the nodal spectral-element method, level-set function-based immersed-interface method and Floquet stability analysis. The different wake states are categorised based on the time-instantaneous wake structure, non-dimensional frequency and aerodynamic force coefficients. The wake states and transition boundaries are summarised in a wake regime map. The critical angle of attack and Reynolds number for the supercritical Hopf bifurcation (i.e. steady to periodic wake transition) varies as $\alpha _1 {\sim} {\textit {Re}}^{-0.65}$, while the critical angle of attack for the onset of three dimensionality varies as $\alpha _{3D} {\sim} {\textit {Re}}^{-0.5}$. Over the entire Reynolds number range, the transition to 3-D flow occurs through a mode C (subharmonic) transition. Beyond this initial transition, further instabilities of the 2-D periodic base flow arise and are investigated. For instance, at $ {\textit {Re}}=2000$ and $\alpha _{3D,2}=11.0^\circ$, mode C coexists together with modes related to modes A and QP seen in a stationary circular cylinder wake. In contrast, at $ {\textit {Re}}=5000$ and $\alpha _{3D,2}=8.0^\circ$, the dominant mode C coexists with mode QP. Three-dimensional simulations well beyond critical angles indicate that 2-D vortex-street transitions are approximately maintained in the fully saturated 3-D wakes in a spanwise-averaged sense.

Buoyant material such as microplastics accumulate near the ocean surface in regions with convergent surface currents where they can be harmful to marine life. Here, we use large eddy simulations to investigate the transport and accumulation of buoyant material in a turbulent ocean mixed layer under combined wind and convection forcing. We model non-inertial buoyant particles with a combination of buoyant tracers and Lagrangian surface particles, which allows us to explore a wide range of particle buoyancies. Surface cooling drives convection, and under this regime persistent convective vortices form that trap buoyant particles, leading to large concentrations. Despite their small size, the convective vortices exhibit a bias towards cyclonic vorticity that has not been reported previously. Based on an analysis of Lagrangian trajectories, the average time that a particle spends inside a convective vortex is long enough for planetary vorticity to become important and further vortex stretching causes an exponential increase in vorticity. When wind forcing is included, there is a transition from convective cells to longitudinal wind rolls with three distinct flow patterns observed under weak, moderate and strong wind forcing. For sufficiently weak winds, convective vortices survive but are less effective at trapping buoyant material. Under strong wind forcing, convective vortices no longer exist, but some clustering occurs in regions of high speed associated with longitudinal wind rolls. We quantify the degree of clustering using the Gini coefficient and find that clustering is strongly influenced by the relative size of the friction and convective velocities and the particle buoyancy.

Instantaneous features of three-dimensional velocity fields are most directly visualized via streamsurfaces. It is generally unclear, however, which streamsurfaces one should pick for this purpose, given that infinitely many such surfaces pass through each point of the flow domain. Exceptions to this rule are vector fields with a non-degenerate first integral whose level surfaces globally define a continuous, one-parameter family of streamsurfaces. While generic vector fields have no first integrals, their vortical regions may admit local first integrals over a discrete set of streamtubes, as Hamiltonian systems are known to do over Cantor sets of invariant tori. Here we introduce a method to construct such first integrals approximately from velocity data, and show that their level sets indeed frame vortical features of the velocity field in examples in which those features are known from Lagrangian analysis. Moreover, we test our method in numerical datasets, including a flow inside a V-junction and a turbulent channel flow. For the latter, we propound an algorithm to pin down the most salient barriers to momentum transport up to a given scale providing a way out of the occlusion conundrum that typically accompanies other vortex visualization methods.

We perform direct numerical simulations of a gas bubble dissolving in a surrounding liquid. The bubble volume is reduced due to dissolution of the gas, with the numerical implementation of an immersed boundary method, coupling the gas diffusion and the Navier–Stokes equations. The methods are validated against planar and spherical geometries’ analytical moving boundary problems, including the classic Epstein–Plesset problem. Considering a bubble rising in a quiescent liquid, we show that the mass transfer coefficient $k_L$ can be described by the classic Levich formula $k_L = (2/\sqrt {{\rm \pi} })\sqrt {\mathscr {D}_l\,U(t)/d(t)}$, with $d(t)$ and $U(t)$ the time-varying bubble size and rise velocity, and $\mathscr {D}_l$ the gas diffusivity in the liquid. Next, we investigate the dissolution and gas transfer of a bubble in homogeneous and isotropic turbulence flow, extending Farsoiya et al. (J. Fluid Mech., vol. 920, 2021, A34). We show that with a bubble size initially within the turbulent inertial subrange, the mass transfer coefficient in turbulence $k_L$ is controlled by the smallest scales of the flow, the Kolmogorov $\eta$ and Batchelor $\eta _B$ microscales, and is independent of the bubble size. This leads to the non-dimensional transfer rate ${Sh}=k_L L^\star /\mathscr {D}_l$ scaling as ${Sh}/{Sc}^{1/2} \propto {Re}^{3/4}$, where ${Re}$ is the macroscale Reynolds number ${Re} = u_{rms}L^\star /\nu _l$, with $u_{rms}$ the velocity fluctuations, $L^*$ the integral length scale, $\nu _l$ the liquid viscosity, and ${Sc}=\nu _l/\mathscr {D}_l$ the Schmidt number. This scaling can be expressed in terms of the turbulence dissipation rate $\epsilon$ as ${k_L}\propto {Sc}^{-1/2} (\epsilon \nu _l)^{1/4}$, in agreement with the model proposed by Lamont & Scott (AIChE J., vol. 16, issue 4, 1970, pp. 513–519) and corresponding to the high $Re$ regime from Theofanous et al. (Intl J. Heat Mass Transfer, vol. 19, issue 6, 1976, pp. 613–624).

We calculate the net energy per unit time exchanged between two sets of modes in a generic system governed by a three-wave kinetic equation. Our calculation is based on the property of detailed energy conservation of the triadic resonant interactions. In a first application to isotropic systems, we re-derive the previously used formula for the energy flux as a particular case for adjacent sets. We then exploit the new formalism to quantify the level of locality of the energy transfers in the example of surface capillary waves. A second application to anisotropic wave systems expands the currently available set of tools to investigate magnitude and direction of the energy fluxes in these systems. We illustrate the use of the formalism by characterizing the energy pathways in the oceanic internal wavefield. Our proposed approach, unlike traditional approaches, is not limited to stationarity, scale invariance and strict locality. In addition, we define a number $w$ that quantifies the scale separation necessary for two sets of modes to having negligible mutual energy exchange, with potential consequences in the interpretation of wave turbulence experiments. The methodology presented here provides a general, simple and systematic approach to energy fluxes in wave turbulence.

We present a family of exact inviscid three-dimensional Beltrami flows in a horizontally periodic domain lying between two parallel free-slip boundaries. Significantly, these flows are not stress free: the horizontal vorticity varies on each boundary. Using direct numerical simulations (employing horizontal hyperdiffusion only for numerical stability), we find that the largest-scale member of the family is unstable and breaks down into anisotropic turbulence, with relatively large horizontal vorticity at and near each boundary, and associated surface frontal features. We conjecture that all members of the family are similarly unstable. The free-slip boundaries play an important role in the late stages of the instability by constraining the deformation of vortex lines near the boundaries. This study appears to be the first to consider the role of boundary horizontal vorticity in an inviscid context.

Atmospheric entry capsules shaped as spherically blunted, large apex-angle cones are widely used in space missions. In Part 1 of this study (Hornung, Martinez Schramm & Hannemann, J. Fluid Mech., vol. 871, 2019, pp. 1097–1116) we explored flows over the two elements of the capsule shape, the sphere and the sharp cone with detached shock, theoretically and computationally. Using a large number of inviscid, perfect-gas computations, analytical functions of two independent variables, the normal-shock density ratio $\varepsilon$ and a cone-angle parameter $\eta$ (which is a function of $\varepsilon$ and the cone half-angle $\theta$) were found for the dimensionless shock wave stand-off distance and the drag coefficient of a sharp cone. An analytical description was found for the shock stand-off distance in the transition from the 90$^\circ$ cone (flat-faced cylinder) to the sphere. In Part 1, it was speculated that the perfect-gas results have relevance to non-equilibrium situations if the normal-shock density ratio is replaced by the density ratio based on the average density along the stagnation streamline. In Part 2, the investigation is extended to blunted-cone capsule shapes. High-precision force measurements and schlieren image analysis are performed in the High-Enthalpy Shock Tunnel Göttingen (HEG) of the German Aerospace Centre using air as the test gas, at conditions where vibrational non-equilibrium effects are significant. Accordingly, results are compared with viscous numerical predictions using different physico-chemical models. A theoretical model is constructed for the density profile along the stagnation streamline that is determined by the free stream conditions and gives the average density. Comparisons of the experimental and numerical results for the dimensionless shock stand-off distance and the drag coefficient, with the extension of the analytical functions of Part 1 to vibrationally relaxing flow, exhibit very good agreement in all of a range of geometries.

The dynamics of axial flow is investigated in initially two-dimensional (2-D) concentrated line vortices that are subsequently cut by a thin wall. The vortices are formed in the wake of an airfoil oscillating sinusoidally at small amplitude, and their circulation and peak vorticity are controlled by the choice of oscillation frequency. The flow field is quantified in terms of the spatio-temporal evolution of axial (spanwise) velocity using one-component molecular tagging velocimetry, and simultaneous measurements of spanwise vorticity and axial velocity at selected planes away from the cutting wall using stereoscopic molecular tagging velocimetry. Results show that the characteristics of the vortices cut by the wall are initially in good agreement with the baseline 2-D vortices. However, as the axial flow develops near the cutting wall and propagates away from it farther downstream, the structure of the initially 2-D Gaussian-shaped vortices quickly changes to a highly distorted region(s) of vorticity near the wall. Data over planes one and two core radii away from the wall reveal appreciable axial velocity both toward and away from the wall, whereas at locations farther away the axial velocity is primarily away from the wall. The data reveal the existence of small amplitude area-varying waves that propagate away from the wall along the vortex core. These disturbances are associated with local changes in the vortex axial velocity and peak vorticity. The propagation speed of these disturbances is found to agree with predictions using the analytical model of Lundgren & Ashurst (J. Fluid Mech., vol. 200, 1989, pp. 283–307).

Piezoelectric macro-fibre composite (MFC) actuators are employed onto the flexible leeward surface of an airfoil for active control. Time-resolved aerodynamic forces, membrane deformations and flow fields are synchronously measured at low Reynolds number (Re = 6 × 104). Mean aerodynamics show that the actively controlled airfoil can achieve lift-enhancement and drag-reduction simultaneously in the angle of attack range of 10° ≤ α ≤ 14°, where the rigid airfoil encounters stall. The maximum increments of lift and lift-to-drag ratio are 27.1 % and 126 % at the reduced actuation frequency of ${f^ + } = 3.52$. The unsteady coupling features are further analysed at α = 12°, where the maximum lift-enhancement occurs. It is newly discovered that the membrane vibrations and flow fields are locked into half of the actuation frequency when ${f^ + } > 3$. The shift of the dominant vibration mode from bending to inclining is the reason for the novel ‘half-frequency lock-in’ phenomenon. To the fluid–structure interaction, there are three characteristic frequencies for the actively controlled airfoil: $S{t_1} = 0.5{f^ + }$, $S{t_2} = {f^ + }$, and $S{t_3} = 1.5{f^ + }$. Here, St1 and its harmonics (St2, St3) are coupled with the natural frequencies of the leading-edge shear layer, resulting in the generation of multi-scale flow structures and suppression of flow separation. The lift presents comparable dominant frequencies between St1 and St3, which means the instantaneous lift is determined by the flow structures of St1 and St3. The local membrane bulge and dent affect the instantaneous swirl strength of flow structures near the maximum vibration amplitude location, which is the main reason for the variation of instantaneous lift.