We consider a layer of an inviscid fluid with a free surface which is subject to vertical high-frequency vibrations. We derive three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency) free-surface waves. The first set of equations is obtained without assuming that the waves are long. These equations are as difficult to solve as the exact equations for irrotational water waves in a non-vibrating fluid. The other two models describe long waves. These models are obtained under two different assumptions about the amplitude of the vibration. Surprisingly, the governing equations have exactly the same form in both cases (up to the interpretation of some constants). These equations reduce to the standard dispersionless shallow-water equations if the vibration is absent, and the vibration manifests itself via an additional term which makes the equations dispersive and, for small-amplitude waves, is similar to the term that would appear if surface tension were taken into account. We show that our dispersive shallow-water equations have both solitary and periodic travelling wave solutions and discuss an analogy between these solutions and travelling capillary–gravity waves in a non-vibrating fluid.

A model for particle propulsion by an instantaneous heat discharge is presented. The flow is driven by a gravity-independent transient fluid dilatation, engendered by an unsteady temperature field which corresponds to heat emission from a localized source located within the particle. We focus on the highly eccentric case, where the heat is released in proximity to the particle surface. Solution of the Stokes equations and subsequent evaluation of the resulting hydrodynamic thrust yields a nonlinear non-autonomous ordinary differential equation governing the evolution of particle position with time. This equation depends upon a single parameter which represents the relative effects of heating magnitude and initial geometry.

Experiments have shown that when a mixture of fuel and oxygen is passed through a zirconia tube whose inner surface is coated with a catalyst, and then ignited at the end of the tube, a reaction front, or flame, propagates back along the tube towards the fuel inlet. The reaction front is visible as a red hot region moving at a speed of a few millimetres per second. In this paper we study a model of the flow, which takes into account diffusion, advection and chemical reaction at the inner surface of the tube. By assuming that the flame propagates at a constant speed without change of form, we can formulate a steady problem in a frame of reference moving with the reaction front. This is solved using the method of matched asymptotic expansions, assuming that the Reynolds and Damköhler numbers are large. We present numerical and, where possible, analytical results, first when the change in fluid density is small (a simplistic but informative limit) and secondly in the variable-density case. The speed of the travelling wave decreases as the critical temperature of the surface reaction increases and as the mass flow rate of fuel increases. We also make a comparison between our results and some preliminary experiments.

We study the inviscid linear stability of a vertical interface separating two fluids of different densities and subject to a gravitational acceleration field parallel to the interface. In this arrangement, the two free streams are constantly accelerated, which means that the linear stability analysis is not amenable to Fourier or Laplace solution in time. Instead, we derive the equations analytically by the initial-value problem method and express the solution in terms of the well-known parabolic cylinder function. The results, which can be classified as an accelerating Kelvin–Helmholtz configuration, show that even in the presence of surface tension, the interface is unconditionally unstable at all wavemodes. This is a consequence of the ever increasing momentum of the free streams, as gravity accelerates them indefinitely. The instability can be shown to grow as the exponential of a quadratic function of time.

This paper describes an experiment in which a uniformly sheared turbulence was subjected to simultaneous streamwise flow curvature and rotation about the streamwise axis. The distortion of the turbulence is complex but well defined and may serve as a test case for turbulence model development. The uniformly sheared turbulence was developed in a straight wind tunnel and then passed into a curved tunnel section. At the start of the curved section the plane of the mean shear was normal to the plane of curvature so as to create a three-dimensional or ‘out of plane’ curvature configuration. On entering the curved tunnel, the flow developed a streamwise mean vorticity that rotated the mean shear about the tunnel centreline through approximately 70°, so that the shear was nearly in the plane of curvature and oriented so as to have a stabilizing effect on the turbulence. Hot wire measurements of the mean velocity, mean vorticity, mean rate of strain and Reynolds stress anisotropy development along the wind tunnel centreline are reported. The observed effect of the mean shear rotation on the turbulence was to diminish the shear stress in the plane normal to the plane of curvature while generating non-zero values of the shear stress in the plane of curvature. A rotating frame was identified for which the measured mean velocity field took the form of a simple shear flow. The turbulence anisotropy was transformed to this frame to estimate the effects of frame rotation on the structure of sheared turbulence.

A nonlinear stability analysis of the viscous circular Couette flow to axisymmetric finite-amplitude perturbations under axial periodic boundary conditions is developed. The analysis is based on investigating the properties of a reduced Arnol’d energy-Casimir function $\mathscr{A}_{rd}$ of Wang (Phys. Fluids, vol. 2, 2009, 084104). A weighted kinetic energy of the perturbation, which has a form of ${\rm\Delta}\mathscr{A}_{rd}$, the difference between the reduced Arnol’d function and its base flow value, is used as a Lyapunov function. We show that all the inviscid flow effects as well as all the viscous-dependent terms that are related to the flow boundaries vanish. The evolution of ${\rm\Delta}\mathscr{A}_{rd}$ depends only on the viscous effects of the perturbation’s dynamics inside the flow domain. The requirement for the temporal decay of ${\rm\Delta}\mathscr{A}_{rd}$ leads to two novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric perturbations. The linearized version of these conditions for infinitesimally small perturbations recovers the recent linear stability results by Kloosterziel (J. Fluid Mech., vol. 652, 2010, pp. 171–193). By examining the nonlinear stability conditions, we establish a definite operational region of the viscous circular Couette flow that is independent of the fluid viscosity. In this region of operation, the flow is nonlinearly stable in response to perturbations of any size, provided that the initial total circulation function is above a minimum level determined by the operational conditions of the base flow. Comparisons with historical studies show that our results shed light on the experimental measurements of Wendt (Ing.-Arch., vol. 4, 1933, pp. 577–595) and extend the classical nonlinear stability results of Serrin (Arch. Rat. Mech. Anal., vol. 3, 1959, pp. 1–13) and Joseph & Hung (Arch. Rat. Mech. Anal., vol. 44, 1971, pp. 1–22). When the flow is nonlinearly stable and evolves axisymmetrically for all time, then it always decays asymptotically in time to the circular Couette flow determined uniquely by the set-up of the rotating cylinders. Finally, we derive upper-bound estimates on the decay rate of finite-amplitude perturbations for the solid-body rotation flow between two coaxial rotating cylinders and for the circular Couette flow. We demonstrate via numerical simulations that the theoretical upper bound is relevant to the dynamics of various axisymmetric perturbations tested, where it is strictly obeyed. This present study provides new physical insights into a classical flow problem that was studied for many decades.

We reconsider exact solutions to the Navier–Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (J. Atmos. Sci., vol. 15 (1), 1958, p. 108) and is parameterized by an inverse Reynolds number ϵ. Long's attention (and that of many subsequent investigators) was centred upon the ‘quasi-cylindrical’ (QC) case corresponding to ϵ = 0. We show that the limit ϵ → 0 is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the Navier–Stokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by ϵ and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected.

For a stratified incompressible Euler fluid under gravity confined by rigid boundaries, sources of vorticity are classified with the aim of isolating those which are sensitive to the topological configurations of density isopycnals, for both layered and continuous density variations. The simplest case of a two-layer fluid is studied first. This shows explicitly that topological sources of vorticity are present whenever the interface intersects horizontal boundaries. Accordingly, the topological separation of the fluid domain due to the interface–boundary intersections can contribute additional terms to the vorticity balance equation. This phenomenon is reminiscent of Klein’s ‘Kaffeelöffel’ thought-experiment for a homogeneous fluid (Klein, Z. Math. Phys., vol. 59, 1910, pp. 259–262), and it is essentially independent of the vorticity generation induced by the baroclinic term in the bulk of the fluid. In fact, the two-layer case is generalized to show that for the continuously stratified case topological vorticity sources are generically present whenever density varies along horizontal boundaries. The topological sources are expressed explicitly in terms of local contour integrals of the pressure along the intersection curves of isopycnals with domain boundaries, and their effects on vorticity evolution are encoded by an appropriate vector, termed here the ‘topological vorticity’.

In this paper, we study the velocity field at the density interface of a two-layer stratification system when the flow is forced at the mid-depth of the lower layer by the source–sink forcing method. It is known that, in a sufficiently strong linear stratification, the source–sink forcing in certain configurations produces a single-vortex pattern which corresponds to the lowest eigenmode of the Helmholtz equation (Kanda & Linden 2001). Two types of forcing configuration are used for the two-layer experiments: one that leads to a steady single-vortex pattern in a linear stratification, and one that results in an unsteady irregular state. Strong single-vortex patterns appear intermittently for the former configurations despite the absence of stratification at the forcing height. When the single-vortex pattern occurs at the density interface, a similar flow field extends down to the forcing height. The behaviour is explained as the coupling of the resonant eigenmode at the interface with the horizontal component of the forcing jets. The results show that stratification can organise a flow, even though it is forced by an apparently random three-dimensional forcing.

Equations are derived that relate the vertical hydrodynamic force on two- and three-dimensional structures that are floating in a fluid of infinite depth to the far-field dipole coefficient in the velocity potential. By an application of Green’s theorem to the radiation or scattering potential and a suitable test potential, the heave added mass, the heave damping and the vertical exciting force are shown to be expressible in terms of the dipole coefficient in the relevant potential. The results add to the known reciprocity relations, which relate quantities such as the damping and the exciting force to the amplitude of the far-field radiated wave. The expressions are valid at all frequencies, and their high- and low-frequency asymptotics are investigated.

A class of exact solutions of the Navier–Stokes equations is derived. Each of them represents the velocity field v=U+u of a thin vortical layer (a planar jet) under a uniform strain velocity field U in three-dimensional infinite space, and provides a simple flow model in which nonlinear coupling between small eddies plays a key role in small-scale vortex dynamics. The small-scale structure of the velocity field is studied by numerically analysing the Fourier spectrum of u. It is shown that the Fourier spectrum of u falls off exponentially with wavenumber k for large k. The Taylor expansion in powers of the coordinate (say y) in the direction perpendicular to the vortical layer suggests that the solution may be well approximated by a function with certain poles in the complex y-plane. The Fourier spectrum based on the singularities is in good agreement with that obtained numerically, where the exponential decay rate is given by the distance of the poles from the real axis of y.

The time development of small three-dimensional disturbances in plane Poiseuille flow of helium II is considered. The study is conducted by considering the interaction of a normal fluid field and a superfluid field. The interaction is caused by a mutual friction forcing between the two flow fields. Specifically, the stability of the normal fluid affected by the mutual forcing is considered. Compared to the ordinary fluid case where the mutual forcing is not present, the presence of the mutual forcing implies a substantial increase of the transient growth of the disturbances. The increase of the transient growth occurs because the mutual forcing reduces the damping of the disturbances. The phase of transient growth becomes thereby more prolonged and higher levels of amplification are reached. There is also a minor effect on the transient growth caused by the modification of the mean flow owing to the mutual forcing. The strongest transient growth occurs for streamwise elongated disturbances, i.e. disturbances with streamwise wavenumber α = 0. When α increases beyond zero, the transient amplification quickly becomes reduced. Striking differences compared to the ordinary fluid case are that the largest transient amplification does not occur when the spanwise wavenumber (β) is close to two and that the peak level of the disturbance energy density amplification does not depend on the square of the Reynolds number.

An exact Craik–Criminale (CC) solution to the incompressible Navier–Stokes (NS) equations describes the instability of an elliptical columnar flow interacting with a single Kelvin wave. These CC solutions are extended to allow multi-harmonic Kelvin waves to interact with any exact ‘base’ solution of the NS equations. The interaction is evaluated along an arbitrarily chosen flowline of the base solution, so exact nonlinear instability in this context is locally convective, rather than absolute. Furthermore, an iterative method called ‘WKB-bootstrapping’ is introduced which successively adds Kelvin waves with incommensurate phases to the extended CC solutions. In illustrating WKB bootstrapping, we construct a succession of extended nonlinear CC solutions consisting of a circular columnar flow interacting progressively with one (Kelvin 1887), two (Fabijonas & Lifschitz 1996), or three waves with incommensurate phases. The phase of each wave packet is frozen into the previous flow and we examine the exact nonlinear convective instability induced at each stage (primary, secondary, tertiary). At each stage, the flow becomes progressively more unstable.

For the understanding of longshore currents along a natural beach, the effects of bottom unevenness are considered to be important, especially for the flow in the swash zone. Currents in the swash zone are strongly influenced by the bed slope because the effect of gravity overwhelms the effect of the depth change. In the present paper, we investigate these effects and focus on waves propagating from offshore over a flat ocean basin of constant depth to a beach with a sloping wavy bottom. The waves are incident at a small angle to the beach normal, and the bed slope in the alongshore direction is varied slowly. To simplify the problem, only cnoidal waves and solitary waves are considered and the bed level is varied sinusoidally in the longshore direction.

A perturbation method is applied to the two-dimensional nonlinear shallow water equation (two-dimensional NLSWE) for the wave motion in order to generate a more simplified model of wave dynamics consisting of a one-dimensional NLSWE for the direction normal to the beach and an equation for the alongshore direction. The first equation, the one-dimensional NLSWE, is solved by Carrier & Greenspan's transformation. The solution of the second one is found by extending Brocchini & Peregrine's solution for a flat beach. Two methods for the solution of the one- dimensional NLSWE are introduced in order to get a solution applicable to large-amplitude swash motions, where the amplitude is comparable to the beach length. One is the Maclaurin expansion of the solution around the moving shoreline, and the other is Riemann's representation of the solution, which exactly satisfies the one-dimensional NLSWE and the boundary conditions. After doing a consistency check by confirming that Riemann's method, a numerical solution, agrees with the exact solution for an infinitely long, sloping beach, we assumed that the Maclaurin series solution can also describe wave motion in the swash zone properly not only for this model but also for our ‘wavy’, finite beach model.

The solution obtained from the Maclaurin series is then plugged into the equation for the alongshore direction to calculate the shore currents induced by wave run-up and back-wash motions, where a ‘weakly two-dimensional solution’ is derived from geometrical considerations. The results show that since the water depth near the shoreline is comparable to the bed level fluctuations, the flow is strongly affected by the bed unevenness, leading to recognizable changes in shoreline movement and the time-averaged velocity and the mass flux of the flow in the swash zone. More specifically, the inhomogeneity of the alongshore mass flux generates offshore currents because of the continuity condition for the fluid mass.