The generation of a magnetic field in an electrically conducting fluid generally involves the complicated nonlinear interaction of flow turbulence, rotation and field. This dynamo process is of great importance in geophysics, planetary science and astrophysics, since magnetic fields are known to play a key role in the dynamics of these systems. This paper gives an introduction to dynamo theory for the fluid dynamicist. It proceeds by laying the groundwork, introducing the equations and techniques that are at the heart of dynamo theory, before presenting some simple dynamo solutions. The problems currently exercising dynamo theorists are then introduced, along with the attempts to make progress. The paper concludes with the argument that progress in dynamo theory will be made in the future by utilising and advancing some of the current breakthroughs in neutral fluid turbulence such as those in transition, self-sustaining processes, turbulence/mean-flow interaction, statistical and data-driven methods and maintenance and loss of balance.

The transient cavity dynamics during water entry of a heavy, non-rotating sphere impacting a rotating pool of liquid is studied experimentally, numerically and theoretically. We show that the pool rotation advances the transition of the cavity type – from deep seal to surface seal – marked by a reduction in the transitional Froude number. The role of the dimensionless rotational number $\mathcal {S} \equiv \omega R_0/U_0$ on the transient cavity dynamics is unveiled, where $R_0$ is the sphere radius, $\omega$ the angular speed of the liquid and $U_0$ the impact velocity. The rotating background liquid has two discernible effects on the cavity evolution. Firstly, an increase in the underwater pressure field due to centripetal effects; and secondly, a reduction in the pressure of airflow in the cavity neck near the water surface. The non-dimensional pinch-off time of the deep seal shows a robust $1/2$ power-law dependence on the Froude number, but with a reducing prefactor for increasing $\omega$. Our findings reveal that the effects of a rotating background liquid on the water entry can be traced back to the subtle differences in the initial stage splash and the near-surface cavity dynamics.

The motion of a sphere in a viscous gas has been studied since the time of Sir George Gabriel Stokes who explored linear, steady and unsteady flows. While the unsteady Stokes equation is often used to calculate these flows, this continuum treatment cannot capture some key physical phenomena. This includes propulsion of a sphere by temperature gradients on its surface, without convection. Taguchi et al. (J. Fluid Mech., 2021) now calculate the flow generated by the impulsive rotation of a sphere in a gas, a problem first proposed by Stokes, using the linearised Boltzmann-BGK (Bhatnagar, Gross, Krook) equation. This statistical mechanical approach naturally captures continuum through to collisionless flows; the latter occurs even when the gas mean free path is small. The heat flow generated by the sphere is also determined – a non-continuum effect – showing its direction reverses as the flow evolves. The predicted phenomena are yet to be observed in experiment.

A theoretical model is presented for the steady multi-layered flow induced by a plane vertically distributed buoyancy source producing a turbulent wall plume in a ventilated box. While aspects of the stratification and rate of fluid exchange between box and exterior have been studied previously, the streamline pattern and velocity field have not been considered until now, despite having potentially important practical implications for achieving comfort in naturally ventilated buildings and for the indoor spread of airborne contagions. The boundary condition at the wall for each layer is established by deducing the turbulent entrainment rate. Using conformal mapping techniques and Poisson's integral theorem, closed-form solutions for the streamfunction of the induced flow in each layer are established. While the flow near the ceiling was overlooked in the classic model for the multi-layered stratification, after considering the possible flow scenarios, the stratification is re-evaluated herein by incorporating an entraining ceiling current. With a markedly thinner top layer, the refined stratification matches well with the available experimental observations, the restrictions we place on the applicability of the model overcoming the previous over-prediction in the number of interfaces. The magnitude of the dimensionless flow velocity, independent of the wall buoyancy flux and physical scale of the box, decreases significantly with the number of layers. Three types of layer, each with a distinct induced flow pattern, are distinguished and their implications for room airflow considered. Notably, the flow in the base layer represents a continual and smooth flushing of air between the inlet opening and the wall plume, whereas an intermediate layer is almost entirely comprised of near-stagnant air.

Studies on the evaporation of multicomponent droplets have revealed complex and important physical mechanisms, induced by preferential phase change or mediated by external vapour sources, e.g. occurrence of density-driven flows, phase separation, transient Marangoni flow and solutal effects, etc. With the addition of hygroscopic salts, the adhesive property of the droplet can be tuned, and the direction of water vapour mass flux reversed. This paper focuses on the dynamics of hygroscopic aqueous solution droplets, and analyses the interplay between different physical processes. Specifically, a lubrication-type model is established with the assumption of a precursor film in front of the three-phase contact line, which indicates qualitative agreement with our experimental results, quantitatively with respect to the initial spreading rates and qualitatively with respect to the overall behaviour. We derive the expression of absorptive mass flux combining the balance of chemical potential across the solution–air interface and the Hertz–Knudsen equation. Depending on the droplet state and the ambient condition, evaporation or vapour absorption occurs. The evaporative/absorptive mass flux varies both spatially and temporally as the droplet approaches equilibrium. It is demonstrated that the dominating mechanisms, i.e. capillary, thermal Marangoni and solutal Marangoni, compete with each other, and lead to diverse droplet dynamics at different stages of evaporation or vapour absorption. The findings shed light on the physical processes within droplets with both positive and negative interfacial mass fluxes, and provide rational explanations for the experimental observations.

Direct numerical simulations (DNS) are performed to investigate a hypersonic flow over a compression ramp with a free stream Mach number of 7.7 and a free stream Reynolds number of $4.2\times 10^{5}$ based on the flat plate length. The DNS results are validated by comparison with experimental data and theoretical predictions. It is shown that even in the absence of external disturbances, streamwise heat flux streaks form on the ramp surface downstream of reattachment, and that they are non-uniformly distributed in the spanwise direction. The surface heat flux exhibits a low-frequency unsteadiness, which propagates in the streamwise direction. Additionally, the unsteadiness of the heat flux streaks downstream of reattachment is coupled with a pulsation of the reattachment position. By conducting a dynamic mode decomposition (DMD) analysis, several oscillatory modes, characterised by streamwise low-frequency periodicity, are revealed in the separation bubble flow. The DNS results are further explained by a global stability analysis (GSA). Particularly, the flow structure of the leading DMD modes is consistent with that of the oscillatory unstable modes identified by the GSA. It is therefore concluded that the global instabilities are responsible for the unsteadiness of the considered compression ramp flow.

Sedimentation of drops has been widely investigated, although a relatively small number of studies have accounted for the presence of a bounding wall, presumably because of the associated analytical difficulties. In addition, these drops almost always contain impurities in the form of surfactants, which alter the interfacial properties, thereby changing the flow characteristics and the settling dynamics. Therefore, a more physically accurate description of settling should account for both the presence of a bounding wall as well as surfactants, a paradigm that remains poorly addressed. As such, here we analyse the effect of surfactants on a drop settling towards a solid wall in the limit of small deformation. We only account for the interfacial transport of the surface impurities and use bipolar coordinates to represent the fluid motion. Assuming the surfactant transport to be diffusion dominated, asymptotic solutions for the velocity field are derived and are subsequently used to analyse the settling dynamics and deformation of the drop. We show that the surfactants slow down the drop and may either augment or reduce the deformation by a small amount depending on the location of the drop. The effect of surfactant becomes most prominent near the wall, wherein the drop experiences the largest hydrodynamic drag. The changing flow patterns caused by the wall also redistributes the surfactant around the interface, resulting in asymmetric depletion and accumulation near the poles. Our results might have potential significance in areas such as separation processes as well as droplet based microfluidics.

The collisions in a dilute polydisperse suspension of sub-Kolmogorov spheres with negligible inertia settling in a turbulent flow and interacting through hydrodynamics including continuum breakdown on close approach are studied. A statistically significant decrease in ideal collision rate without gravity is resolved via a Lagrangian stochastic velocity-gradient model at Taylor microscale Reynolds number larger than those accessible by current direct numerical simulation capabilities. This arises from the difference between the mean inward velocity and the root-mean-square particle relative velocity. Differential sedimentation, comparable to the turbulent shear relative velocity, but minimally influencing the sampling of the velocity gradient, diminishes the Reynolds number dependence and enhances the ideal collision rate i.e. the rate without interactions. The collision rate is retarded by hydrodynamic interactions between sphere pairs and is governed by non-continuum lubrication as well as full continuum hydrodynamic interactions at larger separations. The collision efficiency (ratio of actual to ideal collision rate) depends on the relative strength of differential sedimentation and turbulent shear, the size ratio of the interacting spheres and the Knudsen number (defined as the ratio of the mean-free path of the gas to the mean radius of the interacting spheres). We develop an analytical approximation to concisely report computed results across the parameter space. This accurate closed form expression could be a critical component in computing the evolution of the size distribution in applications such as water droplets in clouds or commercially valuable products in industrial aggregators.

Combinations of passive and active flow control are used to reduce the aerodynamic drag of a three-dimensional blunt body by manipulating its large-scale wake asymmetries. An Ahmed-like body with a square-back is mounted in ground proximity in the test section of a wind tunnel to produce a canonical turbulent wake at $Re_H = 5 \times 10^5$ based on the height $H$ of the body. By using passive perturbations around the model, the large-scale asymmetry and dynamics of the unforced recirculation region are modified. Depending on the unforced wake equilibrium, additional high-frequency pulsed blowing, coupled with small curved deflecting surfaces along selected edges of the base, produces a very different impact on the drag. On the one hand, forcing the wake along all edges results in important drag reduction of up to 12 % through a wake-shaping mechanism with weak influence on the large-scale asymmetry. On the other hand, the reorganization of the recirculation region equilibrium plays a key role in the observed drag changes when the wake is only forced along some edges of the base. In particular, the symmetrization of the mean wake and the influence of forcing on the interaction mechanism between facing shear layers described by Haffner et al. (J. Fluid Mech., vol. 894, 2020, A14) appears to be one of the main mechanisms involved in drag reduction. Even if asymmetric forcing strategies resulting in symmetrization of the mean wake provide more modest drag reductions of up to 7 % compared with forcing around the whole base, they are more efficient from an energetic point of view. This study provides key ingredients to adapt forcing strategies for drag reduction in the presence of various wake asymmetries typically imposed in real flow conditions around ground vehicles.

A thorough characterization of the shear layer that exists in large eddy simulation data of three separated, compression ramp-generated shock/turbulent boundary layer interactions is presented. Free stream Mach numbers ahead of the separation shock are 2.9, 7.2 and 9.1. The shear layers produced by the separation in these flows have convective Mach numbers of 1.0, 1.9 and 2.0, respectively. It is found that the separation shear layers share many properties associated with canonical compressible mixing layers. A region of approximate similarity is found in each where it is possible to collapse the mean flow profiles into nearly a single similarity profile. Large mixing-layer-like vortical rollers are found in the shear layers and these are shown to become increasingly three-dimensional with increasing convective Mach number. The relation between the peak turbulence stress and the spreading rate was found to be consistent with mixing layer data and mixing layer theory derived from dimensional analysis. Turbulent kinetic energy and Reynolds stress budget analysis revealed that, although the streamwise turbulence production is greater than the canonical mixing layer, the transfer of turbulence energy by the pressure–strain terms and the energy drain by viscosity terms both show similar behaviour to mixing layer data at matching convective Mach number. As a result, the spreading rate and turbulence anisotropy decrease with increasing $M_c$. These conclusions are aided by an accurate and direct measurement of the vortex convection velocity determined from enhanced two-point correlations in the shear flow. The usefulness of studying the shock/turbulent boundary layer flow in this manner is emphasized.

This paper investigates a long-standing question about the effect of surface roughness on turbulent flow: What is the equivalent roughness sand-grain height for a given roughness topography? Deep neural network (DNN) and Gaussian process regression (GPR) machine learning approaches are used to develop a high-fidelity prediction approach of the Nikuradse equivalent sand-grain height $k_s$ for turbulent flows over a wide variety of different rough surfaces. To this end, 45 surface geometries were generated and the flow over them simulated at ${Re}_\tau =1000$ using direct numerical simulations. These surface geometries differed significantly in moments of surface height fluctuations, effective slope, average inclination, porosity and degree of randomness. Thirty of these surfaces were considered fully rough, and they were supplemented with experimental data for fully rough flows over 15 more surfaces available from previous studies. The DNN and GPR methods predicted $k_s$ with an average error of less than 10 % and a maximum error of less than 30 %, which appears to be significantly more accurate than existing prediction formulae. They also identified the surface porosity and the effective slope of roughness in the spanwise direction as important factors in drag prediction.

The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement which characterises these systems motivates the study of leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model, we show that in the constrained geometry of a two-dimensional channel the dynamics is richer than in an unbounded domain: alongside the known regimes of standard leapfrogging and the absence of it, we identify new regimes of image-driven leapfrogging and periodic orbits. Moreover, by solving the Gross–Pitaevskii equation for a Bose–Einstein condensate, we show that all four regimes exist for quantum vortices too. Finally, we discuss the differences between classical and quantum vortex leapfrogging which appear when the quantum healing length becomes significant compared to the vortex separation or the channel size, and when, due to high velocity, compressibility effects in the condensate becomes significant.

We present an experimental investigation of motion onset in simple yield stress fluids. In this context, motion onset refers to the transition from the motionless steady state to a steady flow, as well as the development of motion in a fluid initially at rest. We consider the natural convection of carbopol microgels in a square cavity with differentially heated sidewalls. We use particle image velocimetry and thermometry to reveal the evolution of both temperature and velocity fields. It is a hallmark of yield stress fluids that a critical ratio of the yield stress and buoyancy stresses exists above which the steady state is motionless. We observe this critical behaviour in our experiments. Contrary to the theoretical predictions, however, systematic motion is evident at the onset of all experiments, even when the steady state is motionless. Above the critical limit, extremely slow motion is observed immediately after the onset of the experiment. This is followed by very slow decay to rest, reminiscent of creep behaviour. Below the critical limit, the initial slow dynamics is followed by flow development patterns similar to theoretical predictions based on the Bingham model. We show that motion onset in carbopol microgels is dominated by subyield motion and fluidization, key processes that are not captured by viscoplastic models.

Froth flotation by small air bubbles has been traditionally used in industry to capture fine minerals and other hydrophobic particles. This method, however, is not efficient for capturing very small particles. The present work is motivated by a new agglomeration process that overcomes this lack of efficiency. It consists of mixing a particle suspension and saltwater-filled droplets covered with semi-permeable oil layers. This paper investigates the two-particle dynamics of a solid particle and a semi-permeable spherical drop that expands due to osmosis in an external, pure extensional flow field. A dimensionless engulfment parameter measures the relative effects of droplet growth and convective flow. The computational results from numerical integration determine a transient collision efficiency, which describes the influence of hydrodynamic interactions and osmotic flow on particle capture. The results show that drop expansion, which decays slowly with time, greatly increases particle capture rates, especially for small particles. Moreover, as the engulfment parameter increases, there is a transition from flow-dominated capture to expansion-dominated capture. For the case of a non-expanding droplet, we provide a numerical solution for the transient pair distribution function, which enables us to explain the transient particle-capture rate in terms of the microstructure of the suspension. Furthermore, we derive an analytical expression for the initial collision efficiency at zero times, which agrees with our numerical data. The numerical results for non-expanding droplets at long times show increasing collision efficiency as the permeability increases and when the size ratio is near unity, in agreement with previous steady-state calculations.

A combination of shock tunnel experiments, numerical simulations and theoretical analyses is conducted on V-shaped blunt leading edges (VBLEs) with a wide range of geometric parameters at a free stream Mach number of 6. The interactions between the shock waves induced by the two straight branches and the crotch of the VBLEs set up intriguing wave structures with increases in $R/r$ (i.e. the crotch radius $R$ over the leading edge radius $r$) and $\beta$ (i.e. the half-span angle between the two straight branches), including regular reflection (RR), Mach reflection (MR) and regular reflection from the same family (sRR). These wave structures are observed in shock tunnel experiments and are reproduced by numerical simulations. Of great interest, transitions of shock interactions from RR to MR and from MR to sRR are identified with variation in $R/r$ or $\beta$. It is revealed that the specific geometric constraints of VBLEs, rather than the classic detachment and von Neumann criteria, govern the transitions of shock interactions. From theoretical analyses of the relative geometric positions of the shock structures near the crotch, transition criteria of the shock interactions on VBLEs are established. These theoretical transition criteria achieve good agreement with the numerical and experimental results for a wide range of $R/r$ and $\beta$ and thus show great potential for practical engineering applications, such as the selection of geometric parameters for the cowl lip of a hypersonic inlet.

Turbulent emulsions are complex physical systems characterized by a strong and dynamical coupling between small-scale droplets and large-scale rheology. By using a specifically designed Taylor–Couette shear flow system, we are able to characterize the statistical properties of a turbulent emulsion made of oil droplets dispersed in an ethanol–water continuous solution, at an oil volume fraction up to 40 %. We find that the dependence of the droplet size on the Reynolds number of the flow at a volume fraction of 1 % can be well described by the Hinze criterion. The distribution of droplet sizes is found to follow a log-normal distribution, hinting at a fragmentation process as the possible mechanism dominating droplet formation. Additionally, the effective viscosity of the turbulent emulsion increases with the volume fraction of the dispersed oil phase, and decreases when the shear strength is increased. We find that the dependence of the effective viscosity on the shear rate can be described by the Herschel–Bulkley model, with a flow index monotonically decreasing with increasing oil volume fraction. This finding indicates that the degree of shear thinning systematically increases with the volume fraction of the dispersed phase. The current findings have important implications for bridging the knowledge on turbulence and low-Reynolds-number emulsion flows to turbulent emulsion flows.

We numerically investigated the phenomenon of non-Gaussian normal diffusion of a Brownian colloidal particle in a periodic array of planar counter-rotating convection rolls. At high Péclet numbers, normal diffusion is observed to occur at all times with non-Gaussian transient statistics. This effect vanishes with increasing the observation time. The displacement distributions decay either slower or faster than a Gaussian function, depending on the flow parameters. The sign of their excess kurtosis is related to the difference between two dynamical time scales, namely, the mean exit time of the particle out of a convection roll and its circulation period inside it.

The collapse of a granular column in a liquid is investigated using numerical simulations. From previous experimental studies, it has been established that the dynamics of the collapse is mostly influenced by the Stokes number $St$, comparing grain inertia and viscous fluid dissipation, and the initial volume fraction of the granular column $\phi _i$. However, the full characterization of the collapse in the $(St,\phi _i)$ plane is still missing, restricting its modelling as a physical process for geophysical applications. Only numerical tools can allow the variation over the parameter space $(St,\phi _i)$ that is hardly reachable in experiments as well as a full description of the granular phase that plays a major role in dense granular flows. For this purpose, a dedicated numerical model is used including a discrete element method to resolve the granular phase. The specific objectives of the paper are then twofold: (i) the characterization of the dynamics of the collapse and its final deposit with respect to $(St,\phi _i)$ to complement available experimental data, and (ii) the description of the granular rheology according to these two dimensionless numbers including dilatancy effects. A simple predictive model stems from the obtained results, allowing one to explain the evolution of the final deposit with $(St,\phi _i)$.

We experimentally investigate the effect of geometrical anisotropy for buoyant spheroidal particles rising in a still fluid. All other parameters, such as the Galileo number (the ratio of gravitational to viscous forces) $Ga \approx 6000$, the ratio of the particle to fluid density $\varGamma \approx 0.53$ and the dimensionless moment of inertia $\boldsymbol{\mathsf{I}}^*= \boldsymbol{\mathsf{I}}_p/\boldsymbol{\mathsf{I}}_f$ (with $\boldsymbol{\mathsf{I}}_p$ being the moment of inertia of the particle and $\boldsymbol{\mathsf{I}}_f$ that of the fluid in an equivalent volume), are kept constant. The geometrical aspect ratio of the spheroids, $\chi$ , is varied systematically from $\chi = 0.2$ (oblate) to 5 (prolate). Based on tracking all degrees of particle motion, we identify six regimes characterised by distinct rise dynamics. Firstly, for $0.83 \le \chi \le 1.20$, increased rotational dynamics are observed and the particle flips over semi-regularly in a ‘tumbling’-like motion. Secondly, for oblate particles with $0.29 \le \chi \le 0.75$, planar regular ‘zig–zag’ motion is observed, where the drag coefficient is independent of $\chi$. Thirdly, for the most extreme oblate geometries ($\chi \le 0.25$), a ‘flutter’-like behaviour is found, characterised by precession of the oscillation plane and an increase in the drag coefficient. For prolate geometries, we observed two coexisting oscillation modes that contribute to complex trajectories: the first is related to oscillations of the pointing vector and the second corresponds to a motion perpendicular to the particle's symmetry axis. We identify a ‘longitudinal’ regime ($1.33 \le \chi \le 2.5$), where both modes are active and a different one, the ‘broadside’-regime ($3 \le \chi \le 4$), where only the second mode is present. Remarkably, for the most prolate particles ($\chi = 5$), we observe an entirely different ‘helical’ rise with completely unique features.

A millimetre-size superhydrophobic sphere impacting on the free surface of a quiescent bath can be propelled back into the air by capillary effects and dynamic fluid forces, whilst transferring part of its energy to the fluid. We report the findings of a thorough investigation of this phenomenon, involving different approaches. Over the range from minimum impact velocities required to produce rebounds to impact velocities that cause the sinking of the solid sphere, we focus on the dependence of the coefficient of restitution, contact time and maximum surface deflection on the different physical parameters of the problem. Experiments, simulations and asymptotic analysis reveal trends in the rebound metrics, uncover new phenomena at both ends of the Weber number spectrum, and collapse the data. Direct numerical simulations using a pseudo-solid sphere successfully reproduce experimental data whilst also providing insight into flow quantities that are challenging to determine from experiments. A model based on matching the motion of a perfectly hydrophobic impactor to a linearised fluid free surface is validated against direct numerical simulations and used in the low-Weber-number regime. The hierarchical and cross-validated models in this study allow us to explore the entirety of our target parameter space within a challenging multi-scale system.