While the Taylor–Melcher electrohydrodynamic model entails ionic charge carriers, it addresses neither ionic transport within the liquids nor the formation of diffuse space-charge layers about their common interface. Moreover, as this model is hinged upon the presence of non-zero interfacial-charge density, it appears to be in contradiction with the aggregate electro-neutrality implied by ionic screening. Following a brief synopsis published by Baygents & Saville (Third International Colloquium on Drops and Bubbles, AIP Conference Proceedings, vol. 7, 1989, American Institute of Physics, pp. 7–17) we systematically derive here the macroscale description appropriate for leaky dielectric liquids, starting from the primitive electrokinetic equations and addressing the double limit of thin space-charge layers and strong fields. This derivation is accomplished through the use of matched asymptotic expansions between the narrow space-charge layers adjacent to the interface and the electro-neutral bulk domains, which are homogenized by the strong ionic advection. Electrokinetic transport within the electrical ‘triple layer’ comprising the genuine interface and the adjacent space-charge layers is embodied in effective boundary conditions; these conditions, together with the simplified transport within the bulk domains, constitute the requisite macroscale description. This description essentially coincides with the familiar equations of Melcher & Taylor (Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 111–146). A key quantity in our macroscale description is the ‘apparent’ surface-charge density, provided by the transversely integrated triple-layer microscale charge. At leading order, this density vanishes due to the expected Debye-layer screening; its asymptotic correction provides the ‘interfacial’ surface-charge density appearing in the Taylor–Melcher model. Our unified electrohydrodynamic treatment provides a reinterpretation of both the Taylor–Melcher conductivity-ratio parameter and the electrical Reynolds number. The latter, expressed in terms of fundamental electrokinetic properties, becomes $O(1)$ only for intense applied fields, comparable with the transverse field within the space-charge layers; at this limit the asymptotic scheme collapses. Surface-charge advection is accordingly absent in the macroscale description. Owing to the inevitable presence of (screened) net charge on the genuine interface, the drop also undergoes electrophoretic motion. The associated flow, however, is asymptotically smaller than that corresponding to the Taylor–Melcher circulation. Our successful matching procedure contrasts the analysis of Baygents & Saville, who considered more general electrolytes and were unable to directly match the inner and outer regions. We discuss this difference in detail.

By using and extending earlier results (Liu & Ricca, J. Phys. A, vol. 45, 2012, 205501), we derive the skein relations of the HOMFLYPT polynomial for ideal fluid knots from helicity, thus providing a rigorous proof that the HOMFLYPT polynomial is a new, powerful invariant of topological fluid mechanics. Since this invariant is a two-variable polynomial, the skein relations are derived from two independent equations expressed in terms of writhe and twist contributions. Writhe is given by addition/subtraction of imaginary local paths, and twist by Dehn’s surgery. HOMFLYPT then becomes a function of knot topology and field strength. For illustration we derive explicit expressions for some elementary cases and apply these results to homogeneous vortex tangles. By examining some particular examples we show how numerical implementation of the HOMFLYPT polynomial can provide new insight into fluid-mechanical behaviour of real fluid flows.

We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green–Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green–Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave.

We examine two- and three-dimensional drops steadily sliding down an inclined plate. The contact line of the drop is governed by a model based on the Navier-slip boundary condition and a prescribed value for the contact angle. The drop is thin, so the lubrication approximation can be used. In the three-dimensional case, we also assume that the drop is sufficiently small (its size is smaller than the capillary scale). These assumptions enable us to determine the shape of the drop and derive an asymptotic expression for its velocity. For three-dimensional drops, this expression is matched to a qualitative estimate of Kim et al. (J. Colloid Interface Sci., vol. 247, 2002, pp. 372–380) obtained for arbitrary drops, i.e. not necessarily thin and small. The matching fixes an undetermined coefficient in Kim, Lee and Kang’s estimate, turning it into a quantitative result.

Results of point-particle direct numerical simulations of downward gas–solid flow in smooth and rough vertical channels are presented. Two-way coupling and inter-particle collisions are included. The rough walls are shaped as fixed layers of tiny spherical particles with diameter much smaller than the viscous wall unit. The turbulence attenuation induced by the free solid particles in the gas flow is shown to be enhanced with increasing wall roughness. The so-called feedback force, the force exerted by the free particles on the gas, is decomposed into three contributions: the domain average of the mean feedback force, the non-uniform part of the mean feedback force and the fluctuating part of the feedback force. Since the non-uniformity of the mean feedback force increases with wall roughness, the effect of the non-uniform part of the mean feedback force is investigated in detail. For both smooth and rough walls, the non-uniform part of the mean feedback force is shown to contribute significantly to the particle-induced turbulence attenuation.

This paper presents a mathematical model for predicting the propagation of circumferential waves (acoustic, entropy and vorticity waves) through an annular nozzle. Combustion chambers in modern aero-engines are typically annular, and so a model for circumferential waves is essential for understanding and predicting both combustion noise and thermoacoustic instabilities for such geometries. The linearised Euler equations are solved using the Magnus expansion to obtain the reflection and transmission coefficients of the annular nozzle for acoustic, entropy and vorticity perturbations. Predictions which account for flow physics, such as a non-zero mean flow angle and the generation of vorticity noise, are obtained for the first time. Results are compared with two numerical methods, showing that the mathematical model is able to predict the transmission and reflection of waves for both compact and non-compact frequencies. The model is used to prove one particularly interesting and relevant feature of annular geometries: the generation of a vorticity wave by the acceleration of a circumferential entropy wave. It is shown that this phenomenon originates from the baroclinic torque in the vorticity equation.

We investigate the behaviour of flows, including turbulent flows, driven by a horizontal body force and subject to a vertical magnetic field, with the following question in mind: for a very strong applied magnetic field, is the flow mostly two-dimensional, with remaining weak three-dimensional fluctuations, or does it become exactly 2-D, with no dependence along the vertical direction? We first focus on the quasi-static approximation, i.e. the asymptotic limit of vanishing magnetic Reynolds number, $\mathit{Rm}\ll 1$: we prove that the flow becomes exactly 2-D asymptotically in time, regardless of the initial condition and provided that the interaction parameter $N$ is larger than a threshold value. We call this property absolute two-dimensionalization: the attractor of the system is necessarily a (possibly turbulent) 2-D flow. We then consider the full magnetohydrodynamic (MHD) equations and prove that, for low enough $\mathit{Rm}$ and large enough $N$, the flow becomes exactly 2-D in the long-time limit provided the initial vertically dependent perturbations are infinitesimal. We call this phenomenon linear two-dimensionalization: the (possibly turbulent) 2-D flow is an attractor of the dynamics, but it is not necessarily the only attractor of the system. Some 3-D attractors may also exist and be attained for strong enough initial 3-D perturbations. These results shed some light on the existence of a dissipation anomaly for MHD flows subject to a strong external magnetic field.

Motivated by the importance of small-scale turbulent diapycnal mixing to the closure of the large-scale meridional overturning circulation (MOC) of the oceans, we focus on a model problem which allows us to address the fundamental fluid mechanics that is expected to be characteristic of the oceanographic regime. Our model problem is one in which the initial conditions consist of a stably stratified parallel shear flow which evolves into the turbulent regime through the growth of a Kelvin–Helmholtz wave to finite amplitude followed by transition to turbulence. Through both linear stability analysis and direct numerical simulations (DNS), we investigate the secondary instabilities and the turbulent mixing at a fixed high Reynolds number and for a range of Prandtl numbers. We demonstrate that the oceanographically expected high value of the Prandtl number has a profound influence on the nature of the secondary instabilities that govern the transition process. Specifically through non-separable linear stability analysis, we discover new characteristics for the shear-aligned convective instability such that it is modified into a mixed mode that is driven both by static instability and by shear. The growth rate and ultimate strength of this mode are both strongly enhanced at higher $\mathit{Pr}$ while the growth rate and ultimate strength of the stagnation point instability (SPI), which may compete for control of the transition process, are simultaneously impeded. Of equal importance is the fact that, for higher $\mathit{Pr}$, the characteristic length scales associated with the dominant mixed mode of instability decrease and therefore there ceases to be a strong scale selectivity. In the limit of much higher $\mathit{Pr}$, we conjecture that a wide range of spatial scales become equally unstable so as to support an ‘ultraviolet catastrophe’, in which a direct injection of energy occurs into a broad range of scales simultaneously. We further establish the validity of these analytical results through a series of computationally challenging DNS analyses, and provide a detailed analysis of the efficiency of the turbulent mixing of the density field that occurs subsequent to transition and of the entrainment of fluid into the mixing layer from the high-speed flanks of the shear flow. We show that the mixing efficiency decreases monotonically with increase of the molecular value of the Prandtl number and the expansion of the shear layer is reduced as such entrainment diminishes.

Experiments were conducted to characterize the detonation phase-velocity dependence on charge thickness for two-dimensional detonation in condensed-phase explosive slabs of PBX 9501, PBX 9502 and ANFO. In combination with previous diameter-effect measurements from a cylindrical rate-stick geometry, these data permit examination of the relative scaling of detonation phase velocity between axisymmetric and two-dimensional detonation. We find that the ratio of cylinder radius ($R$) to slab thickness ($T$) at each detonation phase velocity ($D_{0}$) is such that $R(D_{0})/T(D_{0})<1$. The variation in the $R(D_{0})/T(D_{0})$ scaling is investigated with two detonation shock dynamics (DSD) models: a lower-order model relates the normal detonation velocity to local shock curvature, while a higher-order model includes the effect of front acceleration and transverse flow. The experimentally observed $R(D_{0})/T(D_{0})$ (${<}1$) scaling behaviour for PBX 9501 and PBX 9502 is captured by the lower-order DSD theory, revealing that the variation in the scale factor is due to a difference in the slab and axisymmetric components of the curvature along the shock in the cylindrical geometry. The higher-order DSD theory is required to capture the observed $R(D_{0})/T(D_{0})$ (${<}1$) scaling behaviour for ANFO. An asymptotic analysis of the lower-order DSD formulation describes the geometric scaling of the detonation phase velocity between the cylinder and slab geometries as the detonation phase velocity approaches the Chapman–Jouguet value.

We describe a series of experiments in which a constant buoyancy flux $B_{s}$ of salty dyed water of density ${\it\rho}_{s}$ is introduced at the top of a long narrow tank of square cross-section tilted at an angle ${\it\theta}$ to the vertical. The tank is initially filled with fresh clear water of density ${\it\rho}_{0}<{\it\rho}_{s}$, and we investigate the resulting buoyancy-driven turbulent mixing at various tilt angles ${\it\theta}$. Using a light-attenuation image analysis method, we determine the evolution of the reduced gravity $g^{\prime }=g({\it\rho}-{\it\rho}_{0})/{\it\rho}_{0}$ of the mixed fluid in time and space as it propagates towards the bottom of the tank. For all tilt angles tested (${\it\theta}=0^{\circ }$ to ${\it\theta}=45^{\circ }$), we focus exclusively on high-Reynolds-number experiments, where the flow remains turbulent both along the length and across the width of the tank. We find that when ${\it\theta}>0^{\circ }$, the cross-tank component of gravity acts to segregate the dense fluid from the relatively lighter fluid, and a statically stable gradient of $g^{\prime }$ across the width of the tank occurs more frequently than a statically unstable gradient, i.e. $(\partial g^{\prime }/\partial x)<0$ occurs more frequently than $(\partial g^{\prime }/\partial x)>0$. This is in contrast to the case when ${\it\theta}=0^{\circ }$, where instantaneous cross-tank gradients of reduced gravity may be positive or negative, but are equal to zero in an ensemble average. We observe that when ${\it\theta}>0^{\circ }$, the cross-tank gradient of reduced gravity induces a turbulent counterflow where dense fluid flows down the upward-facing surface of the tank and lighter fluid flows in the opposing direction above. We model the evolution of the cross-tank averaged, ensemble averaged reduced gravity $\langle \overline{g^{\prime }}\rangle _{e}$ as a diffusive process using Prandtl’s mixing length theory, building on the model of van Sommeren et al. (J. Fluid Mech., vol. 701, 2012, pp. 278–303) who considered purely vertical tanks. We model the fluctuations (from the cross-tank averaged quantity) of reduced gravity $\langle {\hat{g}}^{\prime }\rangle _{e}$ and counterflow velocity $\langle {\hat{w}}\rangle _{e}$ by characterising the mixing across the width of the tank with a cross-tank turbulent diffusivity ${\it\kappa}_{T,x}$, which we assume is constant in the cross-tank coordinate $x$. We show that the counterflow that exists when ${\it\theta}>0^{\circ }$ acts directly to enhance the effective along-tank turbulent diffusivity ${\it\kappa}_{T,z}$, and from experiments, we find that the mixing length increases approximately linearly with ${\it\theta}$, and that both ${\it\kappa}_{T,x}$ and ${\it\kappa}_{T,z}$ are proportional to $(\partial \langle \overline{g^{\prime }}\rangle _{e}/\partial z)^{1/2}$.

In recent years, the independent measurement of wall shear stress with oil-film or oil-drop interferometry has become a cornerstone of turbulent-boundary-layer research as many arguments depend critically on a precise knowledge of the skin friction ${\it\tau}_{w}^{\ast }$. To our knowledge, all practitioners of oil-drop interferometry have so far used the leading-order similarity solution for asymptotically thin, wedge-shaped, two-dimensional oil films established by Tanner & Blows (J. Phys. E: Sci. Instrum., vol. 9, 1976, pp. 194–202) to relate the evolution of drop thickness to ${\it\tau}_{w}^{\ast }$. It is generally believed that this procedure, if carefully implemented, yields the true time-averaged ${\it\tau}_{w}^{\ast }$ within $\pm 1\,\%$ or possibly better, but the systematic errors due to the finite thickness of the oil film have never been determined. They are analysed here for oil films with a thickness of the order of a viscous unit in a zero-pressure-gradient turbulent boundary layer. Neglecting spanwise surface curvature and surface tension effects, corrections due to the secondary air boundary layer above the oil film are derived with a linearised triple-layer approach that accounts for the turbulent shear-stress perturbation by means of modified van-Driest-type closure models. In addition, the correction due to processing oil drops with a slight streamwise surface curvature as if they were exact wedges is quantified. Both corrections are evaluated for oil-drop interferograms acquired in a zero-pressure-gradient turbulent boundary layer at a Reynolds number of around 3500, based on displacement thickness, and are shown to produce a reduction of the friction velocity relative to the basic Tanner and Blows theory of between $-0.1\,\%$ and $-1.5\,\%$, depending on the mixing-length model. Despite the uncertainty about the true correction, the analysis allows the formulation of some guidelines on where and when to analyse interference fringes in order to minimise the error on the measured wall shear stress.

The dynamic Smagorinsky model for large eddy simulation (LES) of stratified turbulence is studied in this paper. A maximum grid spacing criterion of ${\it\Delta}/L_{b}<0.24$ is found in order to capture several of the key characteristics of stratified turbulence, where ${\it\Delta}$ is the filter scale and $L_{b}$ is the buoyancy scale. These results show that the dynamic Smagorinsky model needs a grid spacing approximately twice as large as the regular Smagorinsky model to reproduce similar results. This improvement on the regular Smagorinsky eddy viscosity approach increases the accuracy of results at small resolved scales while decreasing the computational costs because it allows larger ${\it\Delta}$. In addition, the eddy dissipation spectra in LES of stratified turbulence present anisotropic features, taking energy out of large horizontal but small vertical scales. This trend is not seen in the non-stratified cases, where the subgrid-scale energy transfer is isotropic. Statistics of the dynamic Smagorinsky coefficient $c_{s}$ are investigated; its distribution is peaked around zero, and its standard deviations decrease slightly with increasing stratification. In line with previous findings for unstratified turbulence, regions of increased shear favour smaller $c_{s}$ values; in stratified turbulence, the spatial distribution of the shear, and hence $c_{s}$, is dominated by a layerwise pancake structure. These results show that the dynamic Smagorinsky model presents a promising approach for LES when isotropic buoyancy-scale resolving grids are employed.

A general framework of self-preservation (SP) is established, based on the transport equation of the second-order longitudinal velocity structure function in decaying homogeneous isotropic turbulence (HIT). The analysis introduces the skewness of the longitudinal velocity increment, $S(r,t)$ ($r$ and $t$ are space increment and time), as an SP controlling parameter. The present SP framework allows a critical appraisal of the specific assumptions that have been made in previous SP analyses. It is shown that SP is achieved when $S(r,t)$ varies in a self-similar manner, i.e. $S=c(t){\it\phi}(r/l)$ where $l$ is a scaling length, and $c(t)$ and ${\it\phi}(r/l)$ are dimensionless functions of time and $(r/l)$, respectively. When $c(t)$ is constant, $l$ can be identified with the Kolmogorov length scale ${\it\eta}$, even when the Reynolds number is relatively small. On the other hand, the Taylor microscale ${\it\lambda}$ is a relevant SP length scale only when certain conditions are met. The decay law for the turbulent kinetic energy ($k$) ensuing from the present SP is a generalization of the existing laws and can be expressed as $k\sim (t-t_{0})^{n}+B$, where $B$ is a constant representing the energy of the motions whose scales are excluded from the SP range of scales. When $B=0$, SP is achieved at all scales of motion and ${\it\lambda}$ becomes a relevant scaling length together with ${\it\eta}$. The analysis underlines the relation between the initial conditions and the power-law exponent $n$ and also provides a link between them. In particular, an expression relating $n$ to the initial values of the scaling length and velocity is developed. Finally, the present SP analysis is consistent with both experimental grid turbulence data and the eddy-damped quasi-normal Markovian numerical simulation of decaying HIT by Meldi & Sagaut (J. Turbul., vol. 14, 2013, pp. 24–53).

The interaction of a planar shock wave ($M\approx 1.2$) with an $\text{SF}_{6}$ polygonal inhomogeneity surrounded by air is experimentally investigated. Six polygons including a square, two types of rectangle, two types of triangle, and a diamond are generated by the soap film technique developed in our previous work, in which thin pins are used as angular vertexes to avoid the pressure singularities caused by the surface tension. The evolutions of the shock-accelerated $\text{SF}_{6}$ polygons are captured by a high-speed schlieren system from which wave systems and the interface characteristics can be clearly identified. Both regular and irregular refraction phenomena are observed outside the volume, and more complex wave patterns, including transmitted shock, refracted shock, Mach stem and the interactions between them, are found inside the volume. Two typical irregular refraction phenomena (free precursor refraction, FPR, and free precursor von Neumann refraction, FNR) are observed and analysed, and the transition from FPR to FNR is found, providing the experimental evidence for the transition between different wave patterns numerically found in the literature. Combined with our previous work (Zhai et al., J. Fluid Mech., vol. 757, 2014, pp. 800–816), the reciprocal transitions between FPR and FNR are experimentally confirmed. The velocities and trajectories of the triple points are further measured and it is found that the motions of the triple points are self-similar or pseudo-stationary. Besides the shock dynamics phenomena, the evolutions of these shocked heavy polygonal volumes, which are quite different from the light ones, are captured and found to be closely related to their initial shapes. Specifically, for square and rectangular geometries, the different width–height ratios result in different behaviours of shock–shock interaction inside the volume, and subsequently different features for the outward jet and the interface. Quantitatively, the time-variations of the interface scales, such as the width and the normalized displacements of the edges, are obtained and compared with those from previous work. The comparison illustrates the superiority of the interface formation method and the significant effect of the initial interface shape on the interface features. Furthermore, the characteristics of the vortex core, including the velocity and vortex spacing, are experimentally measured, and the vortex velocity is compared with those from some circulation models to check the validity of the models. The results in the present work enrich understanding of the shock refraction phenomenon and the database of research into Richtmyer–Meshkov instability (RMI).

The kinetic energy balance in Rayleigh–Bénard convection is investigated by means of direct numerical simulations for the Prandtl number range $0.01\leqslant \mathit{Pr}\leqslant 150$ and for fixed Rayleigh number $\mathit{Ra}=5\times 10^{6}$. The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss the profiles of all the terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. By analysing the Prandtl number dependence of the kinetic energy balance, we show that the height dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depths, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sublayers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation.

We describe a fast direct numerical simulation (DNS) method that promises to directly characterise the hydraulic roughness of any given rough surface, from the hydraulically smooth to the fully rough regime. The method circumvents the unfavourable computational cost associated with simulating high-Reynolds-number flows by employing minimal-span channels (Jiménez & Moin, J. Fluid Mech., vol. 225, 1991, pp. 213–240). Proof-of-concept simulations demonstrate that flows in minimal-span channels are sufficient for capturing the downward velocity shift, that is, the Hama roughness function, predicted by flows in full-span channels. We consider two sets of simulations, first with modelled roughness imposed by body forces, and second with explicit roughness described by roughness-conforming grids. Owing to the minimal cost, we are able to conduct direct numerical simulations with increasing roughness Reynolds numbers while maintaining a fixed blockage ratio, as is typical in full-scale applications. The present method promises a practical, fast and accurate tool for characterising hydraulic resistance directly from profilometry data of rough surfaces.

Experiments with a pitching and plunging airfoil are conducted in order to investigate the mechanisms responsible for the formation and detachment of leading edge vortices (LEVs). The chord length is varied from 90 to 180 mm, keeping all other non-dimensional parameters constant, specifically the Reynolds number (17 000), the Strouhal number (0.25), the reduced frequency (0.5) and the effective angle of attack history. It is shown that the mechanism of vortex detachment changes with chord length, evident in a corresponding change in flow topology. One mechanism scales with chord length, the other is attributed to viscous effects in the boundary layer. For the latter mechanism a new scaling of the LEV circulation is introduced. A second experiment investigates the influence of the reduced frequency on the LEV circulation and detachment mechanisms, again keeping all other non-dimensional parameters constant.

The interaction of a single bubble with a single vortex ring in water has been studied experimentally. Measurements of both the bubble dynamics and vorticity dynamics have been done to help understand the two-way coupled problem. The circulation strength of the vortex ring (${\it\Gamma}$) has been systematically varied, while keeping the bubble diameter ($D_{b}$) constant, with the bubble volume to vortex core volume ratio ($V_{R}$) also kept fixed at about 0.1. The other important parameter in the problem is a Weber number based on the vortex ring strength $(\mathit{We}=0.87{\it\rho}({\it\Gamma}/2{\rm\pi}a)^{2}/({\it\sigma}/D_{b});a=\text{vortex core radius},{\it\sigma}=\text{surface tension})$, which is varied over a large range, $\mathit{We}=3{-}406$. The interaction between the bubble and ring for each of the $\mathit{We}$ cases broadly falls into four stages. Stage I is before capture of the bubble by the ring where the bubble is drawn into the low-pressure vortex core, while in stage II the bubble is stretched in the azimuthal direction within the ring and gradually broken up into a number of smaller bubbles. Following this, in stage III the bubble break-up is complete and the resulting smaller bubbles slowly move around the core, and finally in stage IV the bubbles escape. Apart from the effect of the ring on the bubble, the bubble is also shown to significantly affect the vortex ring, especially at low $\mathit{We}$ ($\mathit{We}\sim 3$). In these low-$\mathit{We}$ cases, the convection speed drops significantly compared to the base case without a bubble, while the core appears to fragment with a resultant large decrease in enstrophy by about 50 %. In the higher-$\mathit{We}$ cases ($\mathit{We}>100$), there are some differences in convection speed and enstrophy, but the effects are relatively small. The most dramatic effects of the bubble on the ring are found for thicker core rings at low $\mathit{We}$ ($\mathit{We}\sim 3$) with the vortex ring almost stopping after interacting with the bubble, and the core fragmenting into two parts. The present idealized experiments exhibit many phenomena also seen in bubbly turbulent flows such as reduction in enstrophy, suppression of structures, enhancement of energy at small scales and reduction in energy at large scales. These similarities suggest that results from the present experiments can be helpful in better understanding interactions of bubbles with eddies in turbulent flows.

We study tracer particle transport due to flows created by a self-propelled micro-swimmer, such as a swimming bacterium, alga or a microscopic artificial swimmer. Recent theoretical work has shown that as a swimmer moves in the fluid bulk along an infinite straight path, tracer particles far from its path perform closed loops, whereas those close to the swimmer are entrained by its motion. However, in biologically and technologically important cases tracer transport is significantly altered for swimmers that move in a run-and-tumble fashion with a finite persistence length, and/or in the presence of a free surface or a solid boundary. Here we present a systematic analytical and numerical study exploring the resultant regimes and their crossovers. Our focus is on describing qualitative features of the tracer particle transport and developing quantitative tools for its analysis. Our work is a step towards understanding the ecological effects of flows created by swimming organisms, such as enhanced fluid mixing and biofilm formation.

Particulate flows have mainly been studied under the simplifying assumption of a one-way coupling regime where the disperse phase does not modify the carrier fluid. A more complete view of multiphase flows can be gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange, which is certainly relevant at increasing mass loading. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. The momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere, where the transient disturbance produced by each particle is evaluated in a closed form. The particles are described as lumped point masses, which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the physically relevant interactions between the particles and the fluid which avoids any ‘ad hoc’ assumption. The approach is suited for high-efficiency implementation on massively parallel machines since the transient disturbance produced by the particles is strongly localized in space. We will show that hundreds of thousands of particles can be handled at an affordable computational cost, as demonstrated by a preliminary application to a particle-laden turbulent shear flow.