A numerical investigation of convection-driven dynamos is carried out in the plane layer geometry. Dynamos with different magnetic Prandtl numbers $Pm$ are simulated over a broad range of the Rayleigh number $Ra$. The heat transport, as characterized by the Nusselt number $Nu$, shows an initial departure from the heat transport scaling of non-magnetic Rayleigh–Bénard convection (RBC) as the magnetic field grows in magnitude; as $Ra$ is increased further, the data suggest that $Nu$ grows approximately as $Ra^{2/7}$, but with a smaller prefactor in comparison with RBC. Viscous ($\epsilon _u$) and ohmic ($\epsilon _B$) dissipation contribute approximately equally to $Nu$ at the highest $Ra$ investigated; both ohmic and viscous dissipation approach a Reynolds-number-dependent scaling of the form $Re^a$, where $a \approx 2.8$. The ratio of magnetic to kinetic energy approaches a $Pm$-dependent constant as $Ra$ is increased, with the constant value increasing with $Pm$. The ohmic dissipation length scale depends on $Ra$ in such a way that it is always smaller, and decreases more rapidly with increasing $Ra$, than the viscous dissipation length scale for all investigated values of $Pm$.

The recent numerical study of Barthelemy et al. (J. Fluid Mech., vol. 841, 2018, pp. 463–488) investigated the local properties of two-dimensional (2-D) and three-dimensional (3-D) nonlinear unsteady gravity wave packets in deep and uniform intermediate depth water. Their study focused on the breaking inception transition zone separating maximum recurrence and marginal breaking, and reported that a suitably normalized energy flux localized at the steepest crest in the packet provides a robust breaking threshold parameter. Our present study uses the fully nonlinear boundary integral element method solver developed by Touboul & Kharif (Nat. Haz., vol. 84, issue 2, 2016, pp. 585–598) to investigate breaking inception of 2-D deep water nonlinear water wave packets propagating in the presence of a background current that varies linearly with depth. We seek to validate whether the proposed generic breaking inception threshold holds for the case of constant background vorticity. Results are presented for different packet bandwidths and background vorticity levels.

Using direct numerical simulations, we show that swirling electrovortex flows significantly enhance the mixing of the bottom layer alloy in liquid metal batteries during discharge. By studying the flow in various parameter regimes, we identify and explain a novel scaling law for the intensity of these swirling electrovortex flows. Using this scaling law and the model described in Herreman et al. (Phys. Rev. Fluids, vol. 5, 2020, 074501), we estimate the minimal intensity of the external magnetic field that is needed for the swirling electrovortex to enhance the mixing of the alloys in the bottom electrode of arbitrary liquid metal batteries.

For temperatures near the temperature of maximum density, 3.98 $^{\circ }$C for freshwater, the nonlinearity of the equation of state plays an important role in the density driven dynamics. In this study, we demonstrate that the nonlinear equation of state can lead to large scale differences in the spatial and temporal evolution of freshwater gravity currents when intruding and ambient temperatures are below the temperature of maximum density. The results of this study show that when the temperature of the intruding fluid decreases throughout the evolution of the gravity current, the density difference between the intrusion and the ambient decreases rapidly. When the temperature of intruding fluid increases throughout the evolution of the gravity current, the density difference decreases at a slower rate. The differing rates at which the density difference decreases lead to asymmetries in head location, and vertical extent of intruding fluid, and may have implications for larger scale flows in this temperature regime. These results are robust across the Grashof numbers studied.

Surface waves are excited by mechanical vibration of a cylindrical container having an air/water interface pinned at the rim, and the dynamics of pattern formation is analysed from both an experimental and theoretical perspective. The wave conforms to the geometry of the container and its spatial structure is described by the mode number pair ($n,\ell$) that is identified by long exposure time white light imaging. A laser light system is used to detect the surface wave frequency, which exhibits either a (i) harmonic response for low driving amplitude edge waves or (ii) sub-harmonic response for driving amplitude above the Faraday wave threshold. The first 50 resonant modes are discovered. Control of the meniscus geometry is used to great effect. Specifically, when flat, edge waves are suppressed and only Faraday waves are observed. For a concave meniscus, edge waves are observed and, at higher amplitudes, Faraday waves appear as well, leading to complicated mode mixing. Theoretical predictions for the natural frequency of surface oscillations for an inviscid liquid in a cylindrical container with a pinned contact line are made using the Rayleigh–Ritz procedure and are in excellent agreement with experimental results.

We propose a model and numerical method for the propulsion of rectangular and rhombic lattices of flapping plates at $O$(10–100) Reynolds numbers in incompressible flow. The numerical method uses an adaptive mesh to mitigate singularities at the plates’ edges. We establish convergence rates and find good numerical accuracy in a test problem: Laplace's equation in the region exterior to a plate. We then use the method to establish benchmark results for a single flapping plate, including vortex wake characteristics and Froude efficiency over ranges of flapping amplitude, frequency and Strouhal number. As a prelude to a study of propulsive efficiency in Part 2 (J. Fluid Mech., vol. 915, 2021, A21), we study a key ingredient: the time-averaged input power in lattices of plates. Scaling laws for the mean input power are estimated in the limits of small and large streamwise spacings, using steady flow models with small-gap and Poiseuille-like flows between the plates respectively in the two limits. For both lattice types, the mean input power saturates as the lateral spacing becomes large (and thrust occurs). At small lateral spacings, the rhombic lattices’ input power becomes much larger when the plates overlap. The time-averaged input power in flapping lattices agrees qualitatively with the steady models.

We study propulsion of rectangular and rhombic lattices of flapping plates at $O$(10–100) Reynolds numbers in incompressible flow. The fluid dynamics often converges to time periodic in 5–30 flapping periods, facilitating accurate computations of time-averaged thrust force and input power. We classify the propulsive performances of the lattices and the periodicities of the flows with respect to flapping amplitude and frequency, horizontal and vertical spacings between plates, and oncoming flow velocity. Non-periodic states are most common at small streamwise spacing, large lateral spacing and large Reynolds number. Lattices that are closely spaced in the streamwise direction produce intense vortex dipoles between adjacent plates. The flows transition sharply from drag- to thrust-producing as these dipoles switch from upstream to downstream orientations at critical flow speeds. Near these transitions the flows pass through a variety of periodic and non-periodic states, with and without up–down symmetry, and multiple stable self-propelled speeds can occur. As the streamwise spacing increases (and with large lateral spacing), the plates may shed vortex streets that impinge on downstream neighbours. The most efficient streamwise spacing increases with flapping amplitude. With small lateral spacing, the rectangular lattices have Poiseuille-type flows that yield net drag, while the rhombic lattices may shed vortices and generate net thrust, sometimes with relatively high efficiency. As lateral spacing increases to one plate length and beyond, the rectangular lattices begin to shed vortices and generate thrust, eventually with efficiencies similar to the rhombic lattices', as the two types of flows converge. At $Re = 70$, the lattices’ maximum Froude efficiencies are approximately twice those of an isolated plate (only considering nearly periodic lattice flows). As $Re$ decreases, the lattices’ efficiency advantage increases further.

Selective heating of a microparticle surface had been observed to cause its autonomous movement in a fluid medium due to self-generated temperature gradients. Here, we theoretically investigate the response of such an auto-thermophoretic particle near an isothermal planar wall. We derive an exact solution of the energy equation and employ the Lorentz reciprocal theorem to obtain the translational and rotational swimming velocities in the creeping-flow limit. We report fixed points for vertical movement of the micromotor for its specific orientations relative to the wall. The critical wall gap for fixed points shows unique non-monotonic dependence on the metallic coating coverage on the particle. Also, the micromotor trajectories can be switched either from wall-bound sliding or stationary state to escape from the near-wall zone by tuning the particle and the surrounding fluid pair thermal conductivity contrast. The scenario holds several exclusive distinguishing features from the otherwise extensively studied self-diffusiophoresis phenomenon near an inert wall, despite obvious analogies in the respective constitutive laws relating the fluxes with the gradients of the concerned forcing parameters. The most contrasting locomotion is the ability of a self-thermophoretic micromotor with a large heated cap to migrate towards the wall even if it is initially directed away from the wall. During the stationary states of swimming, the cold portion on the micromotor surface faces away from the wall under all conditions. Such unique aspects hold the potential of being harnessed in practice towards achieving intricate control over the autonomous motion of microparticles in thermally regulated fluidic environments.

The inverse transfer of magnetic helicity is investigated through direct numerical simulations of large-scale mechanically driven turbulent flows in the isothermal ideal magnetohydrodynamics framework. The mechanical forcing is either purely solenoidal or purely compressive and the turbulent statistically stationary states considered exhibit root mean square (RMS) Mach numbers $0.1 \lesssim {\mathcal {M}} \lesssim 11$. A continuous small-scale electromotive forcing injects magnetic helical fluctuations, which lead to the build-up of ever larger magnetic structures. Spectral scaling exponents are observed which, for low Mach numbers, are consistent with previous research done in the incompressible case. Higher compressibility leads to smaller absolute values of the magnetic helicity scaling exponents. The deviations from the incompressible case are comparatively small for solenoidally driven turbulence, even at high Mach numbers, as compared with those for compressively driven turbulence, where strong deviations are already visible at relatively mild RMS Mach numbers ${\mathcal {M}}\gtrsim 3$. Compressible effects can thus play an important role in the inverse transfer of magnetic helicity, especially when the turbulence drivers are rather compressive. Theoretical results observed in the incompressible case can, however, be transferred to supersonic turbulence by an appropriate change of variables, using the Alfvén velocity in place of the magnetic field.

The response of a pressure-induced turbulent separation bubble (TSB) to periodic forcing by pulsed-jet actuators (PJAs) positioned in the upstream boundary layer is investigated experimentally in an attempt to elucidate the mechanism of low-frequency contraction and expansion (‘breathing’) already documented in this flow by Mohammed-Taifour & Weiss (J. Fluid Mech., vol. 799, 2016, pp. 383–412). The TSB is generated on a flat test surface by a combination of adverse and favourable pressure gradients and the free-stream velocity is $25\ \textrm {m}\,\textrm {s}^{-1}$. The results indicate that periodic forcing artificially reduces the size of the TSB by moving separation downstream and reattachment upstream. The smaller TSB is associated with narrower streamwise distributions of average pressure and forward-flow fraction, as well as smaller turbulent stresses in the shear layer bounding the recirculation region. Transient forcing experiments further demonstrate that the TSB responds to upstream forcing with a characteristic time scale that is of the same order of magnitude as that of the breathing motion. Overall, the results of this study support a mechanism whereby the low-frequency breathing motion is a response of the TSB to upstream perturbations that affect the position of separation first and, indirectly, the position of reattachment through a global redistribution of the pressure and velocity fields. The low-frequency behaviour of the TSB appears to be well illustrated by a first-order low-pass filter model that converts the broadband fluctuations of the incoming turbulent boundary layer into a low-frequency, large-scale oscillation of the separation and reattachment fronts, thus leading to a contraction and expansion of the TSB. The results of the continuous forcing experiments also offer new insights into active separation control with PJAs by showing that such actuators generate strong starting vortices that, when convected within an adverse pressure gradient, are associated with a downstream shift of the separation front.

Acoustic radiation pressure acts on fluid inhomogeneities and can drive convection. Inspired by experimental observations of acoustically driven convection inside a spherical plasma bulb, we theoretically investigate the convective modes driven by an acoustic field in a stratified, two-dimensional planar system containing ideal gas and heated from above. In the absence of sound, the system is stable. Gravity determines a threshold acoustic amplitude beyond which part the fluid becomes unstable. Regions of convective stability and instability are separated by the location of the acoustic velocity antinode, and convection results in the regions where the density gradient was not aligned with the gradient in the time-averaged square of the acoustic velocity. At acoustic field levels far beyond threshold, the growth of convective modes with small wavenumbers is suppressed by inertia, while the growth of convective modes with very high wavenumbers is suppressed by viscosity. There is a band of wavenumbers between the two limits with approximately constant growth rate. In the limit of zero gravity and low acoustic fields, the instability grows fastest at a wavenumber that is twice that of the acoustic field. An analogy is drawn between acoustic radiation pressure on inhomogeneities and gravitational forces, which allows us to define an ‘acoustic gravity’ that drives flow approximating thermal convection. We propose that spherical acoustic waves can be used to experimentally study Rayleigh–Bénard convection in a central force, and we lay the theoretical foundations of this approach.

Wall cooling has substantial qualitative and quantitative effects on the development of instabilities and subsequent transition processes in hypersonic boundary layers (HBLs). A sequence of linear stability theory, nonlinear two-dimensional and three-dimensional direct numerical simulations is used to analyse Mach 6 boundary layers, with wall temperatures ranging from near-adiabatic to highly cooled conditions, where the second-mode instability is accompanied by radiation of energy. Decomposition of linear stability modes into their fluid-thermodynamic (acoustic, vortical and thermal) components shows that this radiation comprises both acoustic as well as vortical waves. Furthermore, in these cases, two-dimensional simulations show that the conventional ‘trapped’ nature of second-mode instability is ruptured. A quantitative analysis indicates that although the energy efflux of both acoustic and vortical components increases with wall cooling, the destabilization effect is much stronger and no significant abatement of pressure perturbations is realized. The direct impact of these mechanisms on the transition process itself is examined with high-fidelity simulations of three-dimensional second-mode wavepacket propagation. In the near-adiabatic HBL, the wavepacket remains trapped within the boundary layer and attenuates outside the region of linear instability. However, wavepackets in the cooled wall HBLs amplify and display nonlinear distortion, and transition more rapidly. The structure of the wavepacket also displays different behaviour; moderately cooled walls show bifurcation into a leading turbulent head region and a trailing harmonic region, while highly cooled wall cases display lower convection speeds and significant wavepacket elongation, with intermittent spurts of turbulence in the wake of the head region. This elongation effect is associated with a weakening of the lateral jet mechanism due to the breakdown of spanwise coherent structures. These features have a direct impact on wall loading, including skin friction and heat transfer. In moderately cooled walls, the spatially localized wall loading is similar to those in near-adiabatic walls, with dominant impact due to coherent structures in the leading turbulent head region. In highly cooled walls, the elongated near-wall streaks in the wake region of the wavepacket result in more than twice as large levels of skin friction and heat transfer over a sustained period of time.

The distortion of homogeneous isotropic turbulence interacting with a porous cylinder is calculated by means of the rapid distortion theory (RDT). The porous treatment, characterised by a constant static permeability, is modelled as an impedance boundary condition accounting for the Darcy's flow within the body. The RDT algorithm is first validated through comparisons with published velocity measurements in the stagnation region of an impermeable cylinder placed downstream of a turbulence grid. Subsequently, the impact of porosity on the velocity field is investigated through the analysis of the one-dimensional spectra at different locations near the body and the velocity variance along the stagnation streamline. The porous surface affects the incoming turbulence distortion near the cylinder by reducing the blocking effect of the body and by altering the vorticity deformation caused by the mean flow. The former leads to an attenuation of the one-dimensional velocity spectrum in the low-frequency range, whereas the latter results in an amplification of the high-frequency components. This trend is found to be strongly dependent on the turbulence scale and influences the evolution of the velocity fluctuations in the stagnation region. The porous RDT is finally adapted to model the turbulence distortion in the vicinity of the leading edge of a NACA-0024 profile fitted with melamine foam. The good agreement between the calculations and the experimental results demonstrates that the present methodology can improve the understanding of the physical mechanisms involved in the aerofoil-turbulence interaction noise reduction through porosity and be instrumental in designing such passive noise-mitigation treatments.

Time-resolved particle image velocimetry and hydrogen bubble visualization are used to investigate the wake-induced transition of a 30P30N multi-element airfoil at a fixed angle of attack of 4° within the stowed chord Reynolds-number range of 1.38 × 104 to 3.05 × 104. A special transition routine, strongly affected by the slat wake, is observed in the confluent boundary layer over the 30P30N airfoil. In particular, the effects of slat-wake-triggered double-secondary vortices on the whole transition process are explored in detail. At the initial transition stage, the strong slat-wake disturbances penetrate the boundary layer of the main element and are then amplified by double-exponential growth to generate double-secondary vortices. Compared to the scenarios of simplified geometries (He et al., J. Fluid Mech., vol. 718, 2013, pp. 116–130; He & Wang, Phys. Fluids, vol. 27, 2015, 024106), the double-exponential growth provides stronger fluctuations for the transition. At the intermediate transition stage, the wake disturbances trigger the three-dimensional destabilization of these secondary vortices by direct injection or indirect induction, leading to Λ vortices. The spanwise wavelength of the consequent Λ vortices is therefore locked on by the wake disturbances. At the late transition stage, the Λ vortices evolve into hairpin vortex packets and finally contribute to an attached turbulent boundary layer above the main element. Throughout the transition process, no obvious separation occurs in the mean flow above the main element, revealing potential aerodynamic benefits.

To investigate the effect of atmospheric density and jet-fluid density during supersonic jet-induced cratering on a granular particle bed, large eddy simulation (LES) for the fluid phase coupled with a granular flow model based on kinetic theory is used. Coupling accounts for momentum and energy interaction between particles and fluid. Several simulations are conducted that enable discriminating between the crater characteristics which depend on the atmospheric (or jet) density, and those which depend on the ratio between jet-fluid density and atmospheric density. The crater cross-sectional shapes are controlled by the density ratio between jet fluid and atmosphere. For large such ratios, the craters have a parabolic cross-section shape, are relatively shallow, have large diameters and have subdued ejecta. In contrast, for ratios close to unity, the craters have a conical cross-section, are relatively deep, have relatively small diameters and have substantial ejecta; these craters also display ripples on their walls. The physics leading to these differences is explained. Considerations of particle momentum flux indicate that at same fraction of the crater depth and same radial distance from the jet axis, the radial and vertical components have larger magnitudes for conical craters than for parabolic craters. The spatial distribution of the ratio between the drag force and the force due to the particles resistance to compaction are compared and it is found to be dependent on the jet-to-atmospheric fluid density ratio. A few features of the crater, in particular its depth, only depend on the atmospheric (or jet) fluid density.

We study liquid migration in partly saturated shear bands of granular media where liquid is transported away from the shear-band centre. Earlier studies show that the liquid migration can be modelled as a diffusive process with a shear-rate-dependent diffusion coefficient. Here, we apply this model in a two-dimensional Cartesian split-bottom shear cell with one wide, steady shear band. Initially, a high liquid concentration peak develops at the edges of the shear band, which propagates away from the shear band, splitting the shear cell into a liquid-depleted shear-band region and an outer region not yet affected by the liquid migration. Assuming the liquid transport in the vertical direction is negligible, we simplify the liquid migration model to a one-dimensional form evolving over time. By coordinate transformation, an analytically solvable drift-diffusion model is obtained for liquid migration from the simplified model. From here, we obtain analytical solutions for the liquid concentration as a function of space and time. The significance of the mechanisms is studied in terms of the local Péclet number. While drift enhances drying of the shear band and accumulates the liquid in the peak, diffusion shifts the liquid further away from the shear band. To validate the model, we predict numerically the trajectory of the liquid concentration peak from the continuum model and compare with discrete particle method (DPM) simulations. Our continuum model results give a perfect qualitative and an approximate quantitative agreement with the overall results predicted from the DPM model.

Subglacial lakes are isolated, low-temperature and high-pressure water environments hidden under ice sheets. Here, we use two-dimensional direct numerical simulations in order to investigate the characteristic temperature fluctuations and velocities in freshwater subglacial lakes as functions of the ice overburden pressure, $p_i$, the water depth, $h$, and the geothermal flux, $F$. Geothermal heating is the unique forcing mechanism as we consider a flat ice–water interface. Subglacial lakes are fully convective when $p_i$ is larger than the critical pressure $p_*\approx 2848$ dbar, but self-organize into a lower convective bulk and an upper stably stratified layer when $p_i < p_*$, because of the existence at low pressure of a density maximum at temperature $T_d$ greater than the freezing temperature $T_f$. For both high and low $p_i$, we demonstrate that the Nusselt number, $Nu$, and Reynolds number, $Re$, satisfy classical scaling laws provided that an effective Rayleigh number $Ra_{eff}$ is considered. We show that the convective and stably stratified layers at low pressure are dynamically decoupled at leading order because plume penetration is weak and induces limited entrainment of the stable fluid. From the empirical equation for $Nu$ with $Ra_{eff}$, we derive two sets of closed-form expressions for several variables of interest, including the unknown bottom temperature, in terms of the problem parameters $p_i$, $h$ and $F$. The two predictions correspond to two limiting regimes obtained when the effective thermal expansion coefficient is either approximately constant or linearly proportional to the temperature difference driving the convection.

We consider linear and nonlinear waves in a stratified hydrostatic fluid within a channel of variable area, under the restriction of one-dimensional flow. We derive a modified version of Riemann's invariant that is related to the wave luminosity. This quantity obeys a simple dynamical equation in linear theory, from which the rules of wave reflection are easily discerned; and it is adiabatically conserved in the high-frequency limit. Following a suggestion by Lighthill, we apply the linear adiabatic invariant to predict mildly nonlinear waves. This incurs only moderate error. We find that Lighthill's criterion for shock formation is essentially exact for leading shocks, and for shocks within high-frequency waves. We conclude that approximate invariants can be used to accurately predict the self-distortion of low-amplitude acoustic pulses, as well as the dissipation patterns for weak shocks, in complicated environments such as stellar envelopes. We also identify fully nonlinear solutions for a restricted class of problems.

The development of the flow around a circular cylinder with a smaller diameter control rod in close proximity is the subject of this paper. It has long been known that this is an effective way to attenuate regular vortex shedding leading to reductions in its adverse effects on bluff-body flow. The aim of this study is to improve understanding of the ways the control rod affects the near-wake flow including how it influences the positions of boundary layer separation. Experiments were carried out in a water channel to measure lift and drag forces and particle image velocimetry (PIV) was employed to obtain detailed information on flow structure. The values of important properties were fixed as follows: Reynolds number, 20 000; ratio of cylinder and control rod diameters, 10 : 1; centre-to-centre distance between main cylinder and control rod, 0.7$D$ (where $D$ is the main cylinder diameter). The adjustable parameter was the angular position of the rod, $\theta$, which was varied between $90^{\circ }$ and $180^{\circ }$ from the front stagnation line. Lift and drag forces were measured separately for the main cylinder and the control rod. A new method for identifying flow states is introduced using PIV to interrogate the instantaneous flow velocity in the gap between the main cylinder and the control rod. Similarly to previous studies, three stable flow states were observed together with a bistable state. The bistable state is very sensitive to the control rod angle with a small change of ${\pm }1^{\circ }$ being sufficient to change the flow state.

The onset and physical patterns of gap instability (GI), characterised previously as an inviscid, Kelvin–Helmholtz instability, were investigated experimentally and numerically in an eccentric annular channel with an inner-to-outer diameter ratio $d/D=0.5$ and a length equal to 320 hydraulic diameters. The focus was on the range of low and moderate eccentricities ($0\lesssim e \leqslant 0.5$) and Reynolds numbers ($\textit {Re}\lesssim 12\,000$). It was found that, in laminar flow, GI occurred for $e$ as low as $0.05$. When, however, the flow was turbulent in at least part of the cross-section ($\textit {Re}\gtrsim 5000$), GI remained strong only for $e\geqslant 0.5$ but was essentially undetectable for $e\lesssim 0.3$. For $e$ lower than 0.3, the critical Reynolds number for the onset of GI increased with decreasing $e$. Time-frequency analysis of the velocity time histories has revealed the presence of a single type of dominant mode for $e=0.7$ and all considered $\textit {Re}$, and two distinct, occasionally coexisting, modes for $e \lesssim 0.5$ and $2000 \lesssim \textit {Re} \lesssim 5000$. Within a range of low-$e$ or low-$\textit {Re}$ conditions, quasi-periodic flows were highly intermittent and less energetic in an upstream section of the channel, but became progressively less intermittent further downstream. The energy of such motions generally increased with increasing streamwise distance, e and Re. By exception, this energy decreased with increasing $\textit {Re}$ for $e\leqslant 0.3$ and $\textit {Re}\gtrsim 2000$. The gap vortex street generation mechanism and development was analysed and an improved physical model was proposed.