Mixing and transport of a stratifying scalar are investigated at a density interface imbedded in a turbulent shear flow. Steady-state interfacial shear flows are generated in a laboratory water channel for layer Richardson numbers, Ri, between about 1 and 10. The flow field is made optically homogeneous, enabling the use of laser-induced fluorescence with photodiode array imaging to measure the concentration field at high resolution. False-colour images of the concentration field provide valuable insight into interfacial dynamics: when the local mean shear Richardson number, Ris, is less than about 0.40–0.45, interfacial mixing appears to be dominated by Kelvin–Helmholtz (K–H) instabilities; when Ris is somewhat larger than this, interfacial mixing appears to be dominated by shear-driven wave breaking. In both cases, vertical transport of mixed fluid from the interfacial region into adjacent turbulent layers is accomplished by large-scale turbulent eddies which impinge on the interface and scour fluid from its outer edges.

Motivated by the experimental findings, a model for interfacial mixing and entrainment is developed. A local equilibrium is assumed in which the rate of loss of interfacial fluid by eddy scouring is balanced by the rate of production (local mixing) by interfacial instabilities and molecular diffusion. When a single layer is turbulent and entraining, the model results are as follows: in the molecular-diffusion-dominated regime, δ/h ∼ Pe−1/2 and E ∼ Ri−1Pe−1/2; in the wave-breaking-dominated regime, δ/h ∼ Ri−1/2 and E ∼ Ri−3/2; and in the K–H-dominated regime, δ/h ∼ Ri−1 and E ∼ Ri−2, where δ is the interface thickness, h is the boundary-layer thickness, Pe is the Péclet number, and E is the normalized entrainment velocity. In all three regimes, the maximum concentration anomaly, [gcy ]m ∼ Ri−1. When both layers are turbulent and entraining, E and δ depend on combinations of parameters from both layers.

A numerical method is described, based on the hodograph formulation, for analysing in viscid, free-surface flows over a periodic wall. An efficient implementation of the wall boundary condition results in a straightforward method, accurate for a wide range of bottom undulation heights and flow parameters. It is demonstrated that a series of resonances is possible between the bottom undulations and the free surface. The steady, free-surface profiles are accurately calculated for a wide range of current velocities and are shown to be significantly dimpled by higher harmonics. A study of the flow field indicates that the free-surface shape strongly affects the velocities close to the wall, leading to distributions which change dramatically with current velocity. Some implications of the new results on the phenomena of wall dissolution or material deposition, Bragg scattering of surface waves and sediment transport in rivers, are discussed.

A continuous-flow system is fed with a liquid which turns blue a specified time θ after entering. A more or less stationary pattern is thus set up, those parts of the liquid which have spent times longer than θ in the system being blue while the rest is colourless. The value of θ can be changed at will, enabling details of the flow-pattern to be deduced. The interpretation of the finer points of the colour-pattern is made difficult by the effects molecular interdiffusion of different parts of the liquid.

Double-diffusive lock-exchange gravity currents in the fingering regime are explored via two- and three-dimensional Navier–Stokes simulations in the Boussinesq limit. Even at modest Reynolds numbers, for which single-diffusive gravity currents remain laminar, double-diffusive currents are seen to give rise to pronounced small-scale fingering convection. The front velocity of these currents exhibits a non-monotonic dependence on the diffusivity ratio and the initial stability ratio. Strongly double-diffusive currents lose both heat and salinity more quickly than weakly double-diffusive ones, and they lose salinity more quickly than heat, so that the density difference driving them increases. This differential loss of heat and salinity furthermore results in the emergence of strong local density maxima and minima along the top and bottom walls in the gate region, which in turn promote the formation of secondary, counterflowing currents along the walls. These secondary currents cause the flow to develop a three-layer structure. The late stages of the flow are dominated by currents flowing oppositely to the original ones. Three-dimensional simulation results are consistent with the trends observed in a two-dimensional parametric study. A detailed analysis of the energy budget demonstrates that strongly double-diffusive currents can release several times their initially available potential energy, and convert large amounts of internal energy into mechanical energy via scalar diffusion. Scaling arguments based on the simulation results suggest that even low Reynolds number double-diffusive gravity currents are governed by a balance of buoyancy and turbulent drag.

We study in this work steady laminar flows in a low-density granular gas modelled as a system of identical smooth hard spheres that collide inelastically. The system is excited by shear and temperature sources at the boundaries, which consist of two infinite parallel walls. Thus, the geometry of the system is the same that yields the planar Fourier and Couette flows in standard gases. We show that it is possible to describe the steady granular flows in this system, even at large inelasticities, by means of a (non-Newtonian) hydrodynamic approach. All five types of Couette–Fourier granular flows are systematically described, identifying the different types of hydrodynamic profiles. Excellent agreement is found between our classification of flows and simulation results. Also, we obtain the corresponding nonlinear transport coefficients by following three independent and complementary methods: (i) an analytical solution obtained from Grad’s 13-moment method applied to the inelastic Boltzmann equation; (ii) a numerical solution of the inelastic Boltzmann equation obtained by means of the direct simulation Monte Carlo method; and (iii) event-driven molecular dynamics simulations. We find that, while Grad’s theory does not describe quantitatively well all transport coefficients, the three procedures yield the same general classification of planar Couette–Fourier flows for the granular gas.

We investigate the shape of bubbles in liquid oxygen under magnetic levitation conditions: a magnetic field is applied that polarizes bulk oxygen, and its spatial variation induces a body force opposed to its weight. In these conditions, bubbles appear to have a smooth ellipsoidal shape, which may be prolate (elongated in the vertical direction), oblate (elongated in the horizontal plane) or perfectly spherical. The dependence of the elongation ratio $\eta $ on the volume and levitation position is explored. It is found that the bubble shape is prescribed by the minimization of the sum of surface tension, demagnetization and magnetic–gravitational potential energies.

Pairs of nested vortex rings, one with coils, are evolved numerically to compare their topological numbers to those of recent experiments reported in Scheeler et al. (Science, vol. 357, 2017, pp. 487–491). Included are the twist $Tw$, writhe $Wr$ and self-linking ${\mathcal{L}}_{S}$ numbers, plus centreline helicities ${\mathcal{H}}_{c}$. The questions are: can the experimental numbers be validated and do these numbers have roles in the dynamics of the global helicities ${\mathcal{H}}$ and enstrophies $Z$ with respect to cascades? Topological analysis of the experiments $t=0$ analytic centreline vortex trajectories validates only the writhe measurements, not their values of $Tw$ and ${\mathcal{L}}_{S}$, which obey $Tw\lesssim {\mathcal{L}}_{S}=m\gg Wr$ for $m$-coil rings. Not $Tw\ll Wr$. To suggest why the large twists do not contribute to ${\mathcal{H}}$, it is noted that the mapping of the coiled rings onto the mesh is to a first approximation a single pair of Clebsch potentials, whose self-helicity ${\mathcal{H}}_{S}\equiv 0$. Numerical rings with circulations $\unicode[STIX]{x1D6E4}$, including some single rings, show small initial helicities with ${\mathcal{H}}(0)\approx {\mathcal{H}}_{c}\sim (\text{1 to 2})Wr\unicode[STIX]{x1D6E4}^{2}$$\ll {\mathcal{L}}_{S}\unicode[STIX]{x1D6E4}^{2}$. For time and velocity scales that are consistent with the experiments, as the coils evolve, their $Tw$, $Wr$, ${\mathcal{L}}_{S}$ numbers and their helicities are nearly static until reconnection. Nonetheless, $Wr$ and $Tw$ retain important complementary roles in the dynamics of the global helicity ${\mathcal{H}}$ and enstrophy $Z$, with the evolution of the torsion $\unicode[STIX]{x1D70F}(s)$ profile showing the beginnings of a cascade to small scales.

In this paper the equations for Alfvén waves are examined and conditions necessary for the occurrence of resonance are determined. An experiment using liquid sodium inside a torus is described. Waves were generated by means of alternating current supplied to a coil around the torus. A strong resonance was observed at the fundamental frequency, and a weak resonance at three times this frequency. At the fundamental resonance the alternating magnetic field attained magnitudes more than 9 times the magnitude of the field that would have been generated in free space by the exciting current. The results were in good agreement with prediction.

We study the dynamics of two-phase flows injected into a confined porous layer. A model is derived to describe the evolution of the fluid–fluid interface, where the effective saturation of the injected fluid is zero. The flow is driven by pressure gradients due to injection, the buoyancy due to density contrasts and the interfacial tension between the injected and ambient fluids. The saturation field is then computed after the interface evolution is obtained. The results demonstrate that the flow behaviour evolves from early-time unconfined to late-time confined behaviours. In particular, at early times, the influence of capillary forces drives fluid flow and produces a new self-similar spreading behaviour in the unconfined limit that is distinct from the gravity current solution. At late times, we obtain two new similarity solutions, a modified shock solution and a compound wave solution, in addition to the rarefaction and shock solutions in the sharp-interface limit. A schematic regime diagram is also provided, which summarises all possible similarity solutions and the time transitions between them for the partially saturating flows resulting from fluid injection into a confined porous layer. Three dimensionless control parameters are identified and their influence on the fluid flow is also discussed, including the viscosity ratio, the pore-size distribution and the relative contributions of capillary and buoyancy forces. To underline the relevance of our results, we also briefly describe the implications of the two-phase flow model to the geological storage of $\text{CO}_{2}$, using representative geological parameters from the Sleipner and In Salah sites.

A condition is derived for the hydraulic criticality of a 2-layer flow with transverse variations in both layer velocities and thicknesses. The condition can be expressed in terms of a generalized composite Froude number. The derivation can be extended in order to obtain a critical condition for an N-layer system. The results apply to inviscid flows subject to the usual hydraulic approximation of gradual variations along the channel and is restricted to flows in which the velocity remains single-signed within any given layer. For an intermediate layer with a partial segment of sluggish flow, the long-wave dynamics of the overlying and underlying layers become decoupled.

When a gas bubble rises in a surfactant solution, the velocity field and the distribution of surfactant affect each other. This paper gives the theory for small Reynolds and internal Péclet numbers if the surfactant is gaseous or volatile, if its mass flux across the bubble and around its surface dominates its mass flux through the bulk liquid, and if slowness of both adsorption and convective diffusion must be allowed for.

The theory is tested on the experiments of Kelsall et al. (J. Chem. Soc. Faraday Trans., vol. 92, 1996, p. 3879). Their bubbles rose as expected in a pure liquid until the apparatus was opened to the atmosphere. That significantly slowed the bubbles down. The effect is so sensitive to small concentrations of slowly adsorbing or reacting surfactants that atmospheric carbon dioxide could have caused it, even though it alters the equilibrium surface tension by less than four parts per million in pure air.

There are still unexplained discrepancies between experiment and theory. Additional experiments are suggested that would help to explain them.

The idea of distributing static probe cross-sectional areas so as to render the probe insensitive to Mach number is combined here with that of using non-circular cross-sections to render probes insensitive to yaw and angle of attack. Appropriate non-circular cross-sections are described in detail, and a general means of designing blunt or slender probes to have zero sensitivity to yaw and angle of attack in potential flow is described. Four experimental probes have been tested, and test results are presented. These results show that the probes are quite insensitive to yaw and angle of attack within certain limiting angles, which are assumed to correspond to the onset of flow separation.

Large eddy simulation (LES) is used to study the evolution and structure of a lock-exchange, Boussinesq gravity current forming in a channel partially blocked by a porous layer. This configuration is used to understand how the characteristics of a surface layer containing floating vegetation affects the generation of thermally driven convective water exchange in a long shallow channel. The porous layer, which represents the roots of the floating vegetation, contains a staggered array of rigid square cylinders of edge length $D$ with solid volume fraction $\unicode[STIX]{x1D719}$. The cylinders extend over a depth $h_{1}<H$ below the free surface, where $H$ is the channel depth. The surface current of lighter fluid splits into two layers, one propagating slowly inside the porous layer and the other flowing beneath the porous layer. The main geometrical parameters of the porous layer, $\unicode[STIX]{x1D719}$ and $h_{1}/H$, have a large effect on the dynamics and structure of the surface current and the temporal variation of the front position. For cases with sufficiently large values of $h_{1}/H$ and $\unicode[STIX]{x1D719}$, the front within the porous layer approaches the triangular shape observed for low Reynolds number lock-exchange currents propagating in a channel containing cylinders over its whole volume ($h_{1}/H=1$), and the surface current transitions to a drag-dominated regime in which the front velocity is proportional to $t^{-1/4}$, where $t$ is the time since the current is initiated. For sufficiently high values of $\unicode[STIX]{x1D719}$, the velocity of the fluid inside the porous layer is close to zero at all locations except for those situated close to the lock gate and for some distance behind the front. Close to the front, lighter fluid from below penetrates into the porous layer due to unstable stratification at the bottom of the porous layer. Simulation results are also used to assess how $\unicode[STIX]{x1D719},h_{1}/H$ and the Reynolds number affect the rate at which the heavier fluid situated initially inside the porous layer is removed by the advancing surface current and the main mixing mechanisms. Based on the estimated time scales for flushing the porous (root) layer, we show that flushing can significantly enhance the overall rate of nutrient removal by the floating vegetation by maintaining a higher concentration of nutrients within the root layer.

Mixing is studied in open-flow channels with conformally mapped wavy-wall profiles, using a point-vortex model in two-dimensional irrotational, incompressible mean flow. Unsteady dynamics of the separation bubble induced by oscillatory motion of point vortices located in the trough region produces chaotic mixing in the Lagrangian sense. Significant mass exchange between passive tracer particles inside and outside of the separation bubble forms an efficient mixing region which evolves in size as the vortex moves in the unsteady potential flow. The dynamics closely resembles that obtained by previous authors from numerical solutions of the unsteady Navier–Stokes equations for oscillatory unidirectional flow in a wavy channel. Of the wavy channels considered, the skew-symmetric form is most efficient at promoting passive mixing. Diffusion via gridless random walks increases lateral particle dispersion significantly at the expense of longitudinal particle dispersion due to the opposing effect of mass exchange at the front and rear of the particle ensemble. Active mixing in the wavy channel reveals that the fractal nature of the unstable manifold plays a crucial role in singular enhancement of productivity. Hyperbolic dynamics dominate over non-hyperbolicity which is restricted to the vortex core region. The model is simple yet qualitatively accurate, making it a potential candidate for the study of a wide range of vortex-induced transport and mixing problems.

The generation and energy downscaling of wind-forced long internal waves in strongly stratified small-to-medium sized narrow lakes are studied. A two-layer nonlinear model with forcing and damping is employed. Even though the wave field is fundamentally bidirectional in nature, a domain folding technique is employed to simulate the leading-order internal wave field in terms of a weakly nonlinear weakly dispersive model equation of Korteweg–deVries type. Parametric effects of wind-forcing and environmental conditions, including variable topography and variable basin width, are examined. Energy downscaling from basin-scale waves to shorter scales are quantified by means of a time evolution of the wave energy spectra. It is demonstrated that an internal wave resonance is possible when repetitive wind-forcing events arise with a frequency near the linear seiche frequency. An attempt is made to apply the model to describe the shoaling of long waves on sloping endwall boundaries. Modelling of the energy loss and energy reflection during a shoaling event is calibrated against laboratory experiments.

Brenner's general results for the force and torque experienced by an arbitrary particle simultaneously translating and rotating in a linear shear flow are applied to the spherical cap. The cap's five fundamental resistance tensors are determined and the corresponding tensors for the sphere and circular disk are recovered as special cases. The problem of a freely moving cap in a linear shear is considered and the resulting translational and rotational motions are analysed. The cap's centre of free rotation is found and trajectories of this point are plotted in a few instances. The cap is also found to possess a ‘point of planar motion’ which always moves in a plane perpendicular to the vorticity vector of the undisturbed shear regardless of the initial orientation of the cap. It is shown that the motion of the cap actually serves as a model for the motions of all ‘oblate’ asymmetric bodies of revolution which are moving freely in a linear shear.

The evolution and dynamics of a shallow-water vortex system with high initial Reynolds numbers are investigated experimentally without background rotation. A single vortex is generated by rotating a water mass at the centre of an experimental tank using a bottomless cylinder with internal sectors. The surface velocity field is observed via particle image velocimetry. The experimentally observed vorticity fields indicate that strong shallowness (the ratio of the cylinder diameter to the water depth) and high Reynolds number contribute to the formation of large-scale coherent structures in the form of a tripolar vortex system. The shallow-water vortices with high initial Reynolds numbers experience the transition from turbulent to laminar regimes in their decay process. The proposed first-order vortex decay model predicts that a shallow-water vortex decays as t−1 in the initial turbulent stage and as e−t in the later laminar stage due to horizontal diffusion and bottom friction. The estimated transition time scale from the turbulent to laminar stage increases with initial vortex Reynolds number and with shallowness. By taking the vortex expansion into consideration, the second-order vortex decay model is also presented. The azimuthally ensemble-averaged data elucidate effects of the vortex instabilities and of turbulent energy transfer on the formation of large-scale coherent flow structures. Normal mode analysis of the vortex systems is conducted to study the effect of shallowness and Reynolds number on the generation of two-dimensional large-scale coherent structures. The results show that the perturbation wavenumber of mode 2 is the fastest-growing instability in shallow-water conditions, and its effect depends on initial Reynolds number and shallowness.

A buoyant vortex ring produced by an underwater bursting balloon was studied experimentally. The effect of dimensionless surface tension on characteristics including rise velocity, rate of expansion, circulation, trajectory, and lifetime of the vortex ring bubble was investigated. Results showed reasonable agreement with the literature on vortex rings produced by conventional approaches. It was observed that as the dimensionless surface tension increased, the rise velocity, the circulation and consequently the stability of the vortex ring bubble increased; however, the rate of expansion tends toward constant values. A semi-analytical model is proposed by modifying the drag-based model presented by Sullivan et al. (J. Fluid Mech., vol. 609, 2008, pp. 319–347) to make it applicable to buoyant vortex rings. The modified model suggests that the vortex ring expansion is essentially due to the buoyancy force. An expression is also derived for the circulation in terms of the initial volume of the balloon and the depth at which the balloon bursts.

The local interaction of an oblique shock wave with an unseparated turbulent boundary layer at a shallow two-dimensional compression corner is described by asymptotic expansions for small values of the non-dimensional friction velocity and the flow turning angle. It is assumed that the velocity-defect law and the law of the wall, adapted for compressible flow, provide an asymptotic representation of the mean velocity profile in the undisturbed boundary layer. Analytical solutions for the local mean-velocity and pressure distributions are derived in supersonic, hypersonic and transonic small-disturbance limits, with additional intermediate limits required at distances from the corner that are small in comparison with the boundary-layer thickness. The solutions describe small perturbations in an inviscid rotational flow, and show good agreement with available experimental data in most cases where effects of separation can be neglected. Calculation of the wall shear stress requires solution of the boundary-layer momentum equation in a sublayer which plays the role of a new thinner boundary layer but which is still much thicker than the wall layer. An analytical solution is derived with a mixing-length approximation, and is in qualitative agreement with one set of measured values.