We recently lost valuable colleague and friend, Charlie Doering. The following tribute provides an overview of his life.

We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity $\langle wT \rangle$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac 12$ plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields $\langle w T \rangle \leq \frac {1}{2}(\frac {1}{2}+ \frac {1}{\sqrt {3}} ) - 1.6552\, {\textit {R}}^{-(1/3)}$ uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the $-1/3$ exponent are optimal within our bounding framework. This new result halves the best existing (uniform in $ {\textit {R}}$) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on $ {\textit {R}}$ is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than $\frac 12$ at high $ {\textit {R}}$, which corresponds to the top boundary being hotter than the bottom one on average.

We study experimentally the primary instability of a visco-plastic film flow down an inclined plane. The experimental set-up is a channel with a varying slope angle, in which a permanent flow of a Herschel–Bulkley fluid (carbopol or kaolin) is established. Controlled perturbations are imposed at the entrance of the channel to generate surface waves, and their downstream evolution is observed with a laser sensor system measuring the local fluid thickness. Growth rates and cutoff frequencies are obtained after processing the thickness signal, and experimental critical Reynolds and Bingham numbers are deduced. We find that the experimental stability map obtained is well described by the pseudo-plug model of Balmforth & Liu (J. Fluid Mech., vol, 519, 2004, pp. 33–54), a model obtained after neglecting the film thickness compared with its length. This is not the case for dispersion effects (growth rates, cutoff frequencies, phase speeds), for which a more accurate model is needed.

Shedding of vortices can be observed in the wake of a fixed cylinder at Reynolds numbers larger than $Re=47$. This might give the impression that a vortex-induced vibration (VIV), which occurs when the frequency of vortex shedding in the wake of a flexibly mounted cylinder synchronizes with the natural frequency of the structure, could be observed only at Reynolds numbers larger than $Re=47$. Recent numerical simulations and theoretical work, however, have shown that it is possible to observe VIV at subcritical Reynolds numbers, i.e. Reynolds numbers smaller than $Re=47$. In these studies, a VIV has been observed numerically at Reynolds numbers as low as $Re=22$. In the present work, the first experimental evidence of VIV at subcritical Reynolds number is presented. We have designed and built an experimental set-up that makes it possible to conduct VIV experiments at subcritical Reynolds numbers, and at a constant Reynolds number over the entire lock-in range (i.e. the range for which oscillations are observed). Using this experimental set-up, we have confirmed experimentally that VIV can indeed be observed at subcritical Reynolds numbers, by observing VIV at Reynolds numbers as low as $Re=19$. We have observed subcritical VIV both when the Reynolds number stays constant over the entire lock-in range, and when the Reynolds number increases with increasing reduced velocity, while staying within the subcritical range.

We study the convective and absolute forms of azimuthal magnetorotational instability (AMRI) in a cylindrical Taylor–Couette (TC) flow with an imposed azimuthal magnetic field. We show that the domain of the convective AMRI is wider than that of the absolute AMRI. Actually, it is the absolute instability which is the most relevant and important for magnetic TC flow experiments. The absolute AMRI, unlike the convective one, stays in the device, displaying a sustained growth that can be experimentally detected. We also study the global AMRI in a TC flow of finite height using direct numerical simulation and find that its emerging butterfly-type structure – a spatio-temporal variation in the form of axially upward and downward travelling waves – is in a very good agreement with the linear analysis, which indicates the presence of two dominant absolute AMRI modes in the flow giving rise to this global butterfly pattern.

The ability of microswimmers to deploy optimal propulsion strategies is of paramount importance for their locomotory performance and survival at low Reynolds numbers. Although for perfectly spherical swimmers minimum dissipation requires a neutral-type swimming, any departure from the spherical shape may lead the swimmer to adopt a new propulsion strategy, namely those of puller- or pusher-type swimming. In this study, by using the minimum dissipation theorem for microswimmers, we determine the flow field of an optimal nearly spherical swimmer, and show that indeed depending on the shape profile, the optimal swimmer can be a puller, pusher or neutral. Using an asymptotic approach, we find that amongst all the modes of the shape function, only the third mode determines, to leading order, the swimming type of the optimal swimmer.

Rotations of spheroidal particles immersed in turbulent flows reflect the combined effects of fluid strain and vorticity, as well as the time history of these quantities along the particle's trajectory. Conversely, particle rotation statistics in turbulence provide a way to characterise the Lagrangian properties of velocity gradients. Particle rotations are also important for a range of environmental and industrial processes where particles of various shapes and sizes are immersed in a turbulent flow. In this study, we investigate the rotations of inertialess spheroidal particles that follow Lagrangian fluid trajectories. We perform direct numerical simulations (DNS) of homogeneous isotropic turbulence and investigate the dynamics of different particle shapes at different scales in turbulence using a filtering approach. We find that the mean-square particle angular velocity is nearly independent of particle shape across all scales from the Kolmogorov scale to the integral scale. The particle shape does determine the relative split between different modes of rotation (spinning vs tumbling), but this split is also almost independent of the filter scale suggesting a Lagrangian scale-invariance in velocity gradients. We show how the split between spinning and tumbling can be quantitatively related to the particle's alignment with respect to the fluid vorticity.

Fascinating wake vortex patterns emerge when a circular cylinder is forced to vibrate laterally to a uniform fluid flow, deviating from the well-known Kármán vortex street and first reported by Williamson & Roshko (J. Fluids Struct., vol. 2, 1988, pp. 355–381). The two rows of single vortices (2S mode) can suddenly transition to a row of paired vortices and a row of single vortices (P+S mode) as the forcing amplitude is increased. Further increase in amplitude finds another sudden jump back to the 2S mode. Through a series of elegant and carefully crafted numerical simulations, Matharu et al. (J. Fluid Mech., vol. 918, 2021, A42) determine that the transitions occur via bifurcations, but that underlying these observed ‘jumps’, a continuous evolution of the vortex street between the modes is seen along unstable branches connecting the two modes. As the Reynolds number decreases from 100, bistability and the P+S mode are eventually suppressed.

This paper describes a numerical investigation on the effects of wall temperature and leading-edge bluntness on hypersonic laminar separation induced by a finite-span compression corner. The flow conditions were: Mach number 9.66; Reynolds number $1.34 \times 10^6$ per metre; and stagnation temperature 3150 K. The wall to stagnation temperature ratio ($s_w$) varied from 0.095 to 0.333 and was thus in the subcritical range as per the classification of Brown et al. (J. Fluid Mech., vol. 220, 1990, pp. 309–337). Two leading-edge bluntnesses of $40\ \mathrm {\mu }\textrm {m}$ and $200\ \mathrm {\mu }\textrm {m}$ were used in the investigation. Numerical solutions were obtained using a compressible Navier–Stokes solver and compared with triple-deck solutions obtained using the numerical method of Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285). Separation was induced by ramp angles of $10^{\circ }$ and $20^{\circ }$, which produced near incipient and large separations. The scaled angles, which increased with wall to stagnation temperature ratio, were not sufficient to induce secondary separation in the main recirculation region. Two regimes of shock interference were identified depending on the wall temperature ratio. The corner instability in the form of a stationary wave-packet identified by Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285) for scaled angles $\alpha \geq 3.9$ was investigated but is shown to be a numerical artefact of the algorithm rather than having any physical basis. Increasing both the wall temperature ratio and blunting increased the separation length. And there is an equivalence between cooling the wall and reducing bluntness both leading to a reduced separation length.

When a rising bubble in a Newtonian liquid reaches the liquid–air interface, it can burst, leading to the formation of capillary waves and a jet on the surface. Here, we numerically study this phenomenon in a yield-stress fluid. We show how viscoplasticity controls the fate of these capillary waves and their interaction at the bottom of the cavity. Unlike Newtonian liquids, the free surface converges to a non-flat final equilibrium shape once the driving stresses inside the pool fall below the yield stress. Details of the dynamics, including flow energy budgets, are discussed. The work culminates in a regime map with four main regimes, all with different characteristic behaviours.

The tendency of turbulent flows to produce fine-scale motions from large-scale energy injection is often viewed as a scale-wise cascade of kinetic energy driven by vorticity stretching. This has been recently evaluated by an exact, spatially local relationship (Johnson, P.L. Phys. Rev. Lett., vol. 124, 2020, p. 104501), which also highlights the contribution of strain self-amplification. In this paper, the role of these two mechanisms is explored in more detail. Vorticity stretching and strain amplification interactions between velocity gradients filtered at the same scale account for approximately half of the energy cascade rate, directly connecting the restricted Euler dynamics to the energy cascade. Multiscale strain amplification and vorticity stretching are equally important, however, and more closely resemble eddy viscosity physics. Moreover, ensuing evidence of a power-law decay of energy transfer contributions from disparate scales supports the notion of an energy cascade, albeit a ‘leaky’ one. Besides vorticity stretching and strain self-amplification, a third mechanism of energy transfer is introduced and related to the vortex thinning mechanism important for the inverse cascade in two dimensions. Simulation results indicate this mechanism also provides a net source of backscatter in three-dimensional turbulence, in the range of scales associated with the bottleneck effect. Taken together, these results provide a rich set of implications for large-eddy simulation modelling.

We show that the Stokes–Darcy system, which governs flows through adjacent porous and pure-fluid domains in the two-domain approach without forced filtration, can be recovered from the Helmholtz minimal dissipation principle. While the continuity of normal velocity across the interface is imposed explicitly for mass conservation, only the Beavers–Joseph–Saffman–Jones (BJSJ) interface boundary condition is imposed implicitly, and the balance of the normal-force interface boundary condition appears naturally in the variational process. This set of interface boundary conditions is well-accepted in the mathematics community. We show that these interfacial boundary conditions, at the physically important small-Darcy-number regime, are consistent with continuity of pressure across the interface condition. The tangential velocity and pressure are discontinuous in general but the discontinuity is of the order of the square root of the Darcy number. Hence these interfacial conditions are all approximately consistent in the physically important small-Darcy-number regime. The leading order dynamics in the pure fluid zone is governed by the Stokes system with the no-slip no-penetration boundary condition on the interface between the free zone and porous media at a small Darcy number. The leading order non-trivial dynamics in porous media is governed by the Darcy equation with the pressure on the interface prescribed by the pressure of the leading order Stokes flow in the pure fluid zone. Such a semi-decoupled approach has long been used by the groundwater community. Our result is the first rigorous work quantifying the error of this intuitive approach and relating different interfacial conditions.

The present article investigates the effects of a BZT (Bethe–Zel'dovich–Thompson) dense gas (FC-70) on the development of turbulent compressible mixing layers at three different convective Mach numbers $M_c=0.1$, 1.1 and 2.2. This study extends a previous analysis conducted at $M_c=1.1$ (Vadrot et al., J. Fluid Mech., vol. 893, 2020) Several three-dimensional direct numerical simulations (DNS) of compressible mixing layers are performed with FC-70 using the fifth-order Martin–Hou thermodynamic equation of state (EoS) and air using the perfect gas EoS. After having carefully defined self-similar periods using the temporal evolution of the integrated streamwise production term, the evolutions of the mixing layer growth rate as a function of the convective Mach number are compared between perfect gas and dense gas flows. Results show major differences for the momentum thickness growth rate at $M_c=2.2$. The well-known compressibility-related decrease of the momentum thickness growth rate is reduced in the dense gas. Fluctuating thermodynamics quantities are strongly modified. In particular, temperature variations are suppressed, leading to an almost isothermal evolution. The small scales dynamics is also influenced by dense gas effects, which calls for a specific sub-grid-scale model when computing dense gas flows using large eddy simulation. Additional dense gas DNS are performed at three other initial thermodynamic operating points. DNS performed outside and inside the BZT inversion region do not show major differences. BZT effects themselves therefore only have a small impact on the mixing layer growth.

The breakup of an interface into a cascade of droplets and their subsequent coalescence is a generic problem of central importance to a large number of industrial settings such as mixing, separations and combustion. We study the breakup of a liquid jet introduced through a cylindrical nozzle into a stagnant viscous phase via a hybrid interface-tracking/level-set method to account for the surface tension forces in a three-dimensional Cartesian domain. Numerical solutions are obtained for a range of Reynolds ($Re$) and Weber ($We$) numbers. We find that the interplay between the azimuthal and streamwise vorticity components leads to different interfacial features and flow regimes in $Re$–$We$ space. We show that the streamwise vorticity plays a critical role in the development of the three-dimensional instabilities on the jet surface. In the inertia-controlled regime at high $Re$ and $We$, we expose the details of the spatio-temporal development of the vortical structures affecting the interfacial dynamics. A mushroom-like structure is formed at the leading edge of the jet inducing the generation of a liquid sheet in its interior that undergoes rupture to form droplets. These droplets rotate inside the mushroom structure due to their interaction with the prevailing vortical structures. Additionally, Kelvin–Helmholtz vortices that form near the injection point deform in the streamwise direction to form hairpin vortices, which, in turn, trigger the formation of interfacial lobes in the jet core. The thinning of the lobes induces the creation of holes which expand to form liquid threads that undergo capillary breakup to form droplets.

We examine the effects of horizontally layered heterogeneities on the spreading of two-phase gravity currents in a porous medium, with application to numerous environmental flows, most notably geological carbon sequestration. Heterogeneities, which are ubiquitous within geological reservoirs, affect the large-scale propagation of two-phase flows through the action of small-scale capillary forces, yet the relationship between these small- and large-scale processes is poorly understood. Here, we derive a simple upscaled model for a gravity current under an impermeable cap rock, which we use to investigate the effect of a wide range of centimetre-scale heterogeneities on kilometre-scale plume migration. By parameterising in terms of different types of archetypal layering, we assess the sensitivity of the gravity current to the distribution and magnitude of these heterogeneities. Furthermore, since field measurements of heterogeneities are often sparse or incomplete, we quantify how uncertainty in such measurements manifests as uncertainty in the macroscale flow predictions. Using realistic parameter values, we demonstrate that heterogeneities can enhance plume migration speeds by as much as 200 %, and that uncertainty in field measurements can have dramatic consequences on flow predictions, particularly in post-injection scenarios where the role of capillary forces in heterogeneities is accentuated.

The aim of this study is to establish a fundamental understanding of the flame structure and autoignition characteristics of supercritical hydrothermal flames in three-dimensional shear-driven turbulence. The study involves direct numerical simulation of a non-premixed flame (with fuel comprising a mixture of 10 % $\textrm {H}_2$ and 90 % $\textrm {H}_2\textrm {O}$ in terms of mole fraction) at 25.0 MPa in a slot jet; detailed reaction mechanism and multispecies real-fluid properties are considered in the simulation. Qualitative transient inspection revealed that the flame undergoes a three-stage development process in the streamwise direction: sparse autoignition kernels in the upstream region, intense ignitions and establishment of a continuous flame surface in the middle region, and massive flamelets in the downstream region. Ignition kernels primarily form in the interior of large-scale shear-driven vortices featuring a low scalar dissipation rate. Probability density function (p.d.f.) analysis further confirmed that these kernels mainly form in the premixed combustion mode and on the fuel-lean side, in contrast to the authors’ previous findings concerning autoignition in a two-dimensional mixing layer. Analysis of the preignition chemistry indicator (i.e. $\textrm {H}_2\textrm {O}_{2}$ radicals) revealed that although the fuel-rich condition has a shorter homogeneous autoignition delay time, it does not exhibit any remarkable preignition chemistry or intense heat release in the upstream or middle regions because of its large-scale flow structure. A volume rendering of the dimensionless Damköhler number ($Da$) reveals the distribution of autoignition spots and propagating flames. The joint p.d.f. of the mixture fraction and $Da$ reveals the transition from sparse ignition to intense ignition and, finally, to flame propagation.

We use interface-resolved simulations to study near-wall turbulence modulation by small inertial particles, much denser than the fluid, in dilute/semi-dilute conditions. We considered three bulk solid mass fractions, $\varPsi =0.34\,\%$, $3.37\,\%$ and $33.7\,\%$, with only the latter two showing turbulence modulation. The increase of the drag is strong at $\varPsi =3.37\,\%$, but mild in the densest case. Two distinct regimes of turbulence modulation emerge: for smaller mass fractions, the turbulence statistics are weakly affected and the near-wall particle accumulation increases the drag so the flow appears as a single-phase flow at slightly higher Reynolds number. Conversely, at higher mass fractions, the particles modulate the turbulent dynamics over the entire flow, and the interphase coupling becomes more complex. In this case, fluid Reynolds stresses are attenuated, but the inertial particle dynamics near the wall increases the drag via correlated velocity fluctuations, leading to an overall drag increase. Hence, we conclude that, although particles at high mass fractions reduce the fluid turbulent drag, the solid phase inertial dynamics still increases the overall drag. However, inspection of the streamwise momentum budget in the two-way coupling limit of vanishing volume fraction, but finite mass fraction, indicates that this trend could reverse at even higher particle load.

A new continuum perspective for phoretic motion is developed that is applicable to particles of any shape in ‘microstructured’ fluids such as a suspension of solute or bath particles. Using the reciprocal theorem for Stokes flow it is shown that the local osmotic pressure of the solute adjacent to the phoretic particle generates a thrust force (via a ‘slip’ velocity) which is balanced by the hydrodynamic drag such that there is no net force on the body. For a suspension of passive Brownian bath particles this perspective recovers the classical result for the phoretic velocity owing to an imposed concentration gradient. In a bath of active particles that self-propel with characteristic speed $U_0$ for a time $\tau _R$ and then change direction randomly, taking a step of size $\ell = U_0 \tau _R$, at high activity the phoretic velocity is $\boldsymbol {U} \sim - U_0 \ell \boldsymbol {\nabla } \phi _b$, where $\phi _b$ is a measure of the ‘volume’ fraction of the active bath particles. The phoretic velocity is independent of the size of the phoretic particle and of the viscosity of the suspending fluid. Because active systems are inherently out of equilibrium, phoretic motion can occur even without an imposed concentration gradient. It is shown that at high activity when the run length varies spatially, net phoretic motion results in $\boldsymbol {U} \sim - \phi _b U_0 \boldsymbol {\nabla } \ell$. These two behaviours are special cases of the more general result that phoretic motion arises from a gradient in the swim pressure of active matter. Finally, it is shown that a field that orients (but does not propel) the active particles results in a phoretic velocity $\boldsymbol {U} \sim - \phi _b U_0 \ell \boldsymbol {\nabla }\varPsi$, where $\varPsi$ is the (non-dimensional) potential associated with the field.

This study examines the long-wavelength instabilities of an inviscid parallel time-dependent current on the beta-plane. The basic flow is represented by the Kolmogorov pattern, the amplitude of which is modulated in time. Particular attention is given to the regime in which the corresponding steady flows are stable according to the Rayleigh–Kuo criterion. It is shown that the presence of a fluctuating component, regardless of how weak it may be, always renders the basic current linearly unstable. The destabilization is attributed to the resonant forcing of large-scale Rossby waves. The analysis is based on the asymptotic multiscale model, which is validated by numerical simulations. Since most geophysical flows are inherently time-dependent, the associated shear instabilities could represent a significant and ubiquitous source of barotropic turbulence.

In this work, two-phase flows of Newtonian and/or viscoelastic fluids in a ‘cross-slot’ geometry are investigated both experimentally and numerically in the creeping-flow limit. A series of microfluidic experiments – using Newtonian fluids – have been carried out in different cross-section aspect ratios to support our numerical simulations. The numerical simulations rely on a volume of fluid method and make use of a log-conformation formulation in conjunction with the simplified viscoelastic Phan-Thien and Tanner model. Downstream from the central cross, once the flow has become fully developed, we also estimate analytically the thickness of each fluid layer for both two- and three-dimensional cases. In addition to providing a benchmark test for our numerical solver, these analytical results also provide insight into the role of the viscosity ratio. Injecting two fluids with different elastic properties from each inlet arm is shown to be an effective approach to stabilize the purely elastic instability observed in the cross-slot geometry based on the properties of the fluid with the larger relaxation time. Our results show that interfacial tension can also play an important role in the shape of the interface of the two fluids near the free-stagnation point (i.e. in the central cross). By reducing the interfacial tension force, the interface of the two fluids becomes curved and this can consequently change the curvature of streamlines in this region which, in turn, can modify the purely elastic flow transitions. Thus, increasing interfacial tension is shown to have a stabilizing effect on the associated steady symmetry-breaking purely elastic instability. However, at high values of the viscosity ratio, a new time-dependent purely elastic instability arises most likely due to the change in streamline curvature observed under these conditions. Even when both fluids are Newtonian, outside of the two-dimensional limit, a weak instability arises such that the fluid interface in the depth (neutral) direction no longer remains flat.