The effects of nonlinearity on a train of linear water waves in deep water interacting with underlying currents are investigated numerically via a boundary-integral method. The current is assumed to be two-dimensional and stationary, being induced by a distribution of singularities located beneath the free surface, which impose sharp and gentle surface velocity gradients. For ‘slowly’ varying currents, the fully nonlinear results confirm that opposing currents induce wave steepening and breaking within the region where a high convergence of rays occurs. For ‘rapidly’ varying currents, wave blocking and breaking are more prominent. In this case reflection was observed when sufficiently strong adverse currents are imposed, confirming that at least part of the wave energy that builds up within the caustic can be released in the form of partial reflection and wave breaking. For bichromatic waves, the fully nonlinear results show that partial wave blocking occurs at the individual wave components in the wave groups and that waves become almost monochromatic upstream of the blocking region.

The propagation of a premixed flame inside of a confining vessel filled with combustible fluid is determined using large-activation-energy asymptotics. The flame structure is analysed assuming that spatial and temporal variations in the transverse direction are weak compared to those in the direction normal to the flame surface. The analysis considers weak pressure rise from confinement and also allows for mixtures that are both near and removed from stoichiometry, non-unity reaction orders, temperature-dependent transport coefficients, and general Lewis numbers. The resulting equations for flame propagation speed are expressed in a coordinate-free form and describe the evolution of an arbitrary shaped flame in a general confining flow. These expressions are specifically applied to the case of a spherical flame propagating inside a spherical chamber. The radius at which the confining vessel influences the flame propagation is determined and the various mechanisms influencing flame behaviour are discussed. The results give rise to a simplified asymptotic relationship that provides an improved equation that may be used to more accurately extrapolate unstretched laminar flame speeds from experimental measurements.

Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like and turbulence intensity as , where the Rayleigh number indicates the strength of the driving force (for fixed values of , which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find at . Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with as the critical parameter determining the properties of these modes. Counter-intuitively, in the low- regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.

Previous studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e. velocities are -function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale . A non-zero results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale ) times. The longitudinal (along-flow) shear-induced diffusivity is derived, accurate for all , using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine . The non-dimensionalized depends on time and two parameters: the ratio of Lagrangian to diffusive time scales and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with . At moderate this enhancement is approximately a factor of 3. For classic shear dispersion with , the diffusive time scale determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large , a shear time scale , anticipated by a simple analysis of the particle’s domain-crossing time, determines both the time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time using properties of wall-bounded turbulence.

Taylor–Aris dispersion, the shear-induced enhancement of solute diffusion in the flow direction of the solvent, has been studied intensely in the past half century for the case of steady flow and single-frequency pulsating flows. Here, combining Aris’s method of moments with Dirac’s bra–ket formalism, we derive an expression for the effective solute diffusivity valid for transient Taylor–Aris dispersion in any given time-dependent, multi-frequency solvent flow through straight channels. Our theory shows that the solute dispersion may be greatly enhanced by the time-dependent parts of the flow, and it explicitly reveals how the dispersion coefficient depends on the external driving frequencies of the velocity field and the internal relaxation rates for mass and momentum diffusion. Although applicable to any type of fluid, we restrict the examples of our theory to Newtonian fluids, for which we both recover the known results for steady and single-frequency pulsating flows, and find new, richer structure of the dispersion as function of system parameters in multi-frequency systems. We show that the effective diffusivity is enhanced significantly by those parts of the time-dependent velocity field that have frequencies smaller than the fluid momentum diffusion rate and the solute diffusion rate.

The incompressible Navier–Stokes equations are solved numerically to predict the coupled motion of a falling particle and the surrounding fluid as the particle impacts and rebounds from a planar wall. The method is validated by comparing the numerical simulations of a settling sphere with experimental measurements of the sphere trajectory and the accompanying flow field. The normal collision process is then studied for a range of impact Stokes numbers. A contact model of the liquid–solid interaction and elastic effect is developed that incorporates the elasticity of the solids to permit the rebound trajectory to be simulated accurately. The contact model is applied when the particle is sufficiently close to the wall that it becomes difficult to resolve the thin lubrication layer. The model is calibrated with new measurements of the particle trajectories and reproduces the observed coefficient of restitution over a range of impact Stokes numbers from 1 to 1000.

In the context of an Eulerian fluid description, we investigate the dynamics of a shock wave that is driven by the steady impulsively initiated motion of a two-dimensional planar piston with small corrugations superimposed on its surface. This problem was originally solved by Freeman (Proc. Royal Soc. A, vol. 228, 1955, pp. 341–362), who showed that piston-driven shocks are unconditionally stable when the fluid medium through which they propagate is an ideal gas. Here, we generalize Freeman’s mathematical framework to account for a fluid characterized by an arbitrary equation of state. We find that a sufficient condition for shock stability is , where is the D’yakov parameter and is a critical value less than unity. For values of within this range, linear perturbations imparted to the front by the piston at time attenuate asymptotically as . Outside of this range, the temporal behaviour of perturbations is more difficult to determine and further analysis is required to assess the stability of a shock front under such circumstances. As a benchmark of the main conclusions of this paper, we compare our generalized expression for the linearized shock-ripple amplitude with an independent Bessel-series solution derived by Zaidel’ (J. Appl. Math. Mech., vol. 24, 1960, pp. 316–327) and find excellent agreement.

Using the classical catenary as a motivating example, we use slender-body theory to derive a general theory for thin filaments of arbitrary rheology undergoing large combined stretching and bending, which correctly accounts for the nonlinear geometry of deformation and uses integrated state variables to properly represent the complete deformation state. We test the theory for soft catenaries made of a Maxwell fluid and an elastic yield-stress fluid using a combination of asymptotic and numerical analyses to analyse the dynamics of transient sagging and arrest. We validate our results against three-dimensional finite element simulations of drooping catenaries, and show that our minimal models are easier and faster to solve, can capture all the salient behaviours of the full three-dimensional solution, and provide physical insights into the basic mechanisms involved.

This paper considers the structure and linear stability of two-dimensional hollow vortex equilibria. Equilibrium solutions for a single hollow vortex in linear and nonlinear straining flows are derived in analytical form using free streamline theory. The linear stability properties of this solution class are then determined numerically and a new type of resonance-induced displacement instability is identified. It is found to be a consequence of the fact that one of the shape distortion modes of a circular hollow vortex has the same frequency as one of the modes corresponding to displacement of the vortex centroid. The instability is observed in the case of an isolated hollow vortex situated in straining flow of order three. We also revisit the hollow vortex row solution due to Baker, Saffman & Sheffield (J. Fluid Mech., vol. 74, 1976, p. 1469), and since it is currently lacking in the literature, we present a full linear stability analysis of this solution using Floquet analysis.

Results of three-dimensional numerical simulations of the flow through a sudden expansion in a pipe are presented. The axisymmetric state is known to be stable over the range of Reynolds numbers studied, but recent experimental results suggest bifurcation phenomena. A resolution of this dichotomy between calculation and experiment is provided using imperfections to promote the nonlinear development of asymmetric steady states. These lose stability to disordered motion and the boundary between the steady and time-dependent flows has been established over a range of parameters. Moreover, disordered flows are found to co-exist with the axisymmetric regime when the disturbance is removed from the flow. Hence we provide direct numerical evidence for multiplicity of solutions for the axisymmetric expansion problem, which may have relevance to pipe flows.

Both background rotation and small depths are said to enforce the two-dimensionality of flows. In the current paper, we describe a systematic study of the two-dimensionality of a shallow monopolar vortex subjected to background rotation. Using a perturbation analysis of the Navier–Stokes equations for small aspect ratio (with the fluid depth and a typical radial length scale of the vortex), we found nine different regimes in the parameter space where the flow is governed to lowest order by different sets of equations. From the properties of these sets of equations, it was determined that the flow can be considered as quasi-two-dimensional in only five of the nine regimes. The scaling of the velocity components as given by these sets of equations was compared with results from numerical simulations to find the actual boundaries of the different regimes in the parameter space (), where is the Ekman boundary layer thickness and is the equivalent boundary layer thickness for a monopolar vortex without background rotation. Even though background rotation and small depths do promote the two-dimensionality of flows independently, the combination of these two characteristics does not necessarily have that same effect.

A mechanical representation of batoid-like propulsion using a flexible fin with an elliptical planform shape is used to study the hydrodynamics of undulatory propulsion. The wake is found to consist of a series of interconnected vortex rings, whereby leading and trailing edge vortices of subsequent cycles become entangled with one another. Efficient propulsion is achieved when leading and trailing edge vortices coalesce at the spanwise location where most of the streamwise fluid momentum is concentrated in the wake of the fin. Both the Strouhal number and the wavelength are found to have a significant effect on the wake structure. In general, a decrease in wavelength promotes a wake transition from shedding a single vortex per half-oscillation period to shedding a pair of vortices per half-oscillation period. An increase in Strouhal number causes the wake to bifurcate a finite distance downstream of the trailing edge of the fin into a pair of jets oriented at an acute angle to the line of symmetry. The bifurcation distance decreases with increasing Strouhal number and wavelength, and it is shown to obey a simple scaling law.

In a two-layer density-stratified fluid it is known, due to Ball (J. Fluid Mech., vol. 19, 1964, p. 465), that two oppositely travelling surface waves may form a triad resonance with an interfacial wave. Ball claims ‘there are no other interactions’ between two surface waves and one interfacial wave. Contrary to this, here we present a new class of triad resonance that occurs between two co-propagating surface waves and one interfacial wave. While in Ball’s resonance the interfacial wave has a wavelength of about half of two surface waves, in the new resonance presented here the interfacial wave has a much higher wavelength compared to those of surface waves. This, together with the unidirectionality of the participant triplet, makes the realization of the new resonance more likely in real ocean scenarios. We further show, via theoretical analysis and direct simulation, that, unique to this new class of resonance, the triad inevitably undergoes a cascade of (near-) resonance interaction that spreads the energy of initial waves to a number of lower and higher harmonics. The significance of the resonance studied here is, particularly, more emphasized in the littoral zones, where the spectrum refracts toward a unidirectional wave train.

When a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

In this paper, we investigate the shape stability of a nearly spherical bubble encapsulated by a viscoelastic membrane in an ultrasound field. To describe the dynamic balance on the bubble surface, the in-plane stress and the bending moment are incorporated into the governing equations for the perturbed radial flow of viscous incompressible fluid (Prosperetti, Q. Appl. Math., vol. 34, 1977, p. 339). The radial motion of the bubble is obtained by solving the Rayleigh–Plesset equation with elastic stress. The deflection therefrom is linearized and expanded with respect to the Legendre polynomial of order . Two amplitudes for each shape mode are introduced because the membrane moves not only in the radial direction but also in the tangential direction. The system with a boundary layer approximation is reduced to Mathieu’s equation. A simple expression for the natural frequency of the shape mode is derived, which is validated by direct numerical simulation. Stability diagrams for the higher-order shape mode are mapped out in the phase space of driving amplitude and frequency over a range of values of the elastic modulus of the membrane. The most unstable driving frequency is found to satisfy an integer multiple relationship of the form , due to the structure of Mathieu’s equation in the system. In addition to the resonance interaction, liquid viscosity plays an important role in the stability of the encapsulated bubble.

We study the flow of an electrically conducting fluid past a truncated square cylinder in a rectangular duct under the influence of an externally applied homogeneous magnetic field oriented along the cylinder axis. Our aim is to bridge the gap between the non-magnetic regime, where we previously found a complex set of three-dimensional recirculations behind the cylinder (Dousset & Pothérat, J. Fluid Mech., vol. 653, 2010, pp. 519–536) and the asymptotic regime of dominating Lorentz force analysed by Hunt & Ludford (J. Fluid. Mech., vol. 33, 1968, pp. 693–714). The latter regime is characterized by a remarkable structure known as Hunt’s wake in the magnetohydrodynamics community, where the flow is deflected on either side of a stagnant zone, right above the truncated cylinder as if the latter would span the full height of the duct. In steady flows dominated by the Lorentz force, with negligible inertia, we provide the first numerical flow visualization of Hunt’s wake. In regimes of finite inertia, a thorough topological analysis of the steady flow regimes reveals how the Lorentz force gradually reorganizes the flow structures in the hydrodynamic wake of the cylinder as the Hartmann number (which gives a non-dimensional measure of the magnetic field) is increased. The nature of the vortex shedding follows from this rearrangement of the steady structures by the magnetic field. As is increased, we observe that the vortex street changes from a strongly symmetric one to the alternate procession of counter-rotating vortices typical of the non-truncated cylinder wakes.

This experimental study describes the turbulent wake behind a two-dimensional porous obstruction, consisting of a circular array of cylinders. The cylinders extend from the channel bed through the water surface, mimicking a patch of emergent vegetation. Three patch diameters () and seven solid volume fractions () are tested. Because flow can pass through the patch, directly downstream there is a region of steady, non-zero, streamwise velocity, , called the steady wake. For the patch diameters and solid volume fractions considered here, is a function of only. The length of the steady wake () increases as decreases and can be predicted from the growth of a plane shear layer. The formation of the von-Kármán vortex street is delayed until the end of the steady wake. There are two regions of elevated transverse velocity fluctuation (): directly behind the patch, associated with the wake turbulence of individual cylinders; and at the distance from the patch, associated with the formation of large-scale wake oscillation. Velocity along the centreline of the wake starts to increase only after the patch-scale vortex street is formed, and it approaches the free-stream velocity over a distance . The dimensionless length of the entire wake, , increases with patch porosity.

We present the results of an experimental investigation across a broad range of source Froude numbers, , into the dynamics, morphology and rise heights of Boussinesq turbulent axisymmetric fountains in quiescent uniform environments. Typically, these fountains are thought to rise to an initial height, , before settling back and fluctuating about a lesser (quasi-) steady height, . Our measurements show that this is not always the case and the ratio of the fountain’s initial rise height to steady rise height, , varies widely, , across the range of investigated. As a result of near-ideal start-up conditions provided by the experimental set-up we were consistently able to form a vortex at the fountain’s front. This enabled new insights into two features of the initial rise of turbulent fountains. Firstly, for the initial rise height is less than the steady rise height. Secondly, for , the vortex formed at the fountain’s front pinches off, separates from the main body and rises high above the fountain; there is thus a third rise height to consider, namely, the maximum vortex rise height, . From our observations we propose classifying turbulent axisymmetric fountains into five regimes (as opposed to the current three regimes) and present detailed descriptions of the flow in each. Finally, based on an analysis of the rise height fluctuations and the width of fountains in (quasi-) steady state we provide further insight into the physical cause of height fluctuations.

This paper aims to narrow the gap between the Lagrangian and Eulerian descriptions of rotating stratified fluids. To this end, without loss of generality the primitive Lagrangian equations with arbitrary oriented time-dependent rotation and arbitrary stable stratification have been simplified and made more amenable for analysis. The bulk of the work is concerned with developing in parallel exact Lagrangian and Eulerian descriptions of a particular interesting class of motions of rotating stratified incompressible Boussinesq fluids: the vertically uniform columnar motions. The Lagrangian description is confined to ideal fluids, while the Eulerian one includes viscosity and diffusivity. Assuming the rotation axis to be parallel to gravity, with the rotation rate being an arbitrary function of time, and the buoyancy frequency to be constant, it is found that for vertically uniform motions there is always an exact split into horizontal and vertical subsystems. Evolution of the horizontal velocities and displacements is governed by the classical equations of two-dimensional incompressible hydrodynamics, only slightly modified by accounting for the variable rotation rate. These equations are independent of stratification and vertical motions. The Coriolis term is potential and can be incorporated into pressure. The vertical motions represent a manifestation of packets of inertia–gravity waves with strictly horizontal wavevectors, and are exactly described by linear equations independently of the wave amplitudes. They do not depend on rotation, either constant or variable. The wavepackets do not interact with each other or with horizontal motions. For ideal fluids or those with Rayleigh friction there are explicit solutions describing these motions for arbitrary initial conditions. The Cauchy problem for the columnar motions in ideal fluids is found to be well posed. Thus there is a natural extension of well-studied two-dimensional incompressible hydrodynamics which retains the property of the absence of vortex stretching: all two-dimensional flows could be ‘dressed up’ by adding appropriate vertical motions of a rotating stratified fluid. All the columnar motions could be described in such a way. The examined columnar motions exist under arbitrary relations between the parameters of rotation and stratification and, in particular, without rotation. In the limit of strong rotation one recovers the results known in the literature, in particular, under additional assumptions of small amplitude and steadiness of motions the solutions describe the classical Taylor–Proudman columns.

Inhomogeneous nonlinear shallow-water equations are studied using the Carrier–Greenspan approach and the resulting equations are solved analytically. The Carrier–Greenspan transformations are commonly used hodograph transformations that transform the nonlinear shallow-water equations into a set of linear equations in which partial derivatives with respect to two auxiliary variables appear. Yet, when the resulting initial-value problem is treated analytically through the use of Green’s functions, the partial derivatives of the Green’s functions have non-integrable singularities. This has forced researchers to numerically differentiate the convolutions of the Green’s functions. In this work we remedy this problem by differentiating the initial condition rather than the Green’s function itself; we also perform a change of variables that renders the entire problem more easily treatable. This particular Green’s function approach is especially useful to treat sources that are extended in time; we therefore apply it to model the run-down and run-up of the tsunami waves triggered by submarine landslides. Another advantage of the method presented is that the parametrization of the landslide using sources is done within the integral algorithm that is used for the rest of the problem instead of treating the landslide-generated wave as a separate incident wave. The method proves to be more accurate than the techniques based on Bessel function expansions if the sources are very localized.