The mechanics of the quasi-steady breaking wave created above a submerged hydrofoil, first studied experimentally by Duncan (1981), is elucidated here. It is an example of a flow wherein the resistance of the body manifests itself in a detached separation eddy located away from the body (i.e. on the free surface). As we show, the conditions for inception of separation and the prediction of the breaking configuration follow from simple considerations, without extensive calculation.

The physical model of the breaker, based on observations, consists of an essentially stagnant eddy riding on the forward face of the leading wave in the wave train behind the hydrofoil. This eddy is sustained by turbulent stresses acting in the shear zone separating the eddy and the underlying flow. These stresses result in a trailing turbulent wake just beneath the water surface. The breaker eddy contains air entrained at breaking, and the degree of aeration is a parameter of the problem.

The eddy—breaker model is quantified utilizing independent measurements of turbulent shear stress in shear zones. It is then shown that the hydrostatic pressure acting on the dividing streamline underneath the eddy creates a trailing wave which largely cancels the trailing wave that would exist in the absence of breaking. The ‘wave’ resistance of the hydrofoil then manifests itself in the momentum flux of the residual trailing wave, plus the momentum flux in the breaker wake, i.e. the breaker resistance.

For a fixed hydrofoil speed the total momentum flux, or resistance, in the presence of breaking is shown to have a minimum corresponding to a particular value of the trailing-wave steepness. It is thus concluded that the wave resistance must exceed this value for breaking to ensue. For hydrofoil resistance in excess of this minimum, both a weak and strong breaker would seem to exist. It is shown, however, that the weak breaker is unstable. It is also shown that a maximum steady breaking resistance exists, limited by the size of the breaker and dependent on the extent of its aeration.

Good quantitative comparisons between theory and experiments are shown.

The evolution of spanwise phase variations of nominally two-dimensional instability modes in a plane shear layer is studied in a closed-return water facility using time-harmonic excitation having spanwise-non-uniform phase or frequency distributions. The excitation waveform is synthesized by a linear array of 32 surface film heaters flush-mounted on the flow partition. A span wise-linear phase distribution leads to the excitation of oblique waves and to the rollup of oblique primary vortices. When the prescribed phase distribution is piecewise-constant and spanwise-periodic, the flow is excited with a linear combination of a two-dimensional wavetrain and pairs of equal and opposite oblique waves, the amplitudes of which depend on the magnitude of the phase variation ΔΦ. As a result of the excitation, the primary vortices undergo spanwise-non-uniform rollup and develop spanwise-periodic deformations that induce cross-shear and secondary vortices in the braid region. The amplitude of the deformations of the primary vortices and the shape and strength of the secondary vortices depend on the magnitude of ΔΦ. When ΔΦ is small, the secondary vortices are counter-rotating vortex pairs. As ΔΦ increases, cross-shear induced by oblique segments of the primary vortices in the braid region results in the formation of single secondary vortex strands. The flow is not receptive to spanwise phase variations with wavelengths shorter than the streamwise wavelength of the Kelvin–Helmholtz instability. When the phase variation is ΔΦ = ϕ, the flow is excited with pairs of oblique waves only and undergoes a double rollup, resulting in the formation of spanwise-deformed vortices at twice the excitation frequency. Measurements of the streamwise velocity component show that the excitation leads to a substantial increase in the cross-stream spreading of the shear layer and that distortions of transverse velocity profiles are accompanied by an increase in the high-frequency content of velocity power spectra. Detailed schlieren visualizations shed light on the nature of ‘vortex dislocations’ previously observed by other investigators. Complex spanwise-non-uniform pairing interactions between the spanwise vortices are forced farther downstream by spanwise-amplitude or phase variations of subharmonic excitation wavetrains.

Direct numerical simulations of a fully developed turbulent channel flow for two relatively small values of the Reynolds number are used to examine its influence on various turbulence quantities in the near-wall region. The limiting wall behaviour of these quantities indicates important increases in the r.m.s. value of the wall pressure fluctuations and its derivatives, the r.m.s. streamwise vorticity and in the average energy dissipation rate and the Reynolds shear stress. If the normalization is based on the wall shear stress and the kinematic viscosity, these changes are shown to be consistent with an increase in strength – but not the average diameter or average location – of the quasi-streamwise vortices in the buffer region. Evidence of this strengthening is provided by the increased sum of the stretching terms for the meansquare streamwise vorticity. It is also shown that a normalization based on Kolmogorov velocity and lengthscales, defined at the wall, is more appropriate in the near-wall region than scaling on the wall shear stress and kinematic viscosity.

The method of product integration is applied to the vortex dynamics of two-dimensional incompressible viscous media. In the cases of both unbounded and bounded flows under the no-slip boundary condition, the analytic solutions of the Cauchy problem are obtained for the Helmholtz equation in the form of linear and nonlinear product integrals. The application of product integrals allows the generalization in a natural way of the vortex dynamics concept to the case of viscous flows. However, this new approach requires the reconsideration of some traditional notions of vortex dynamics. Two lengthscales are introduced in the form of a micro- and a macro-scale. Elementary ‘vortex objects’ are defined as two types of singular vortex filaments with equal but opposite intensities. The vorticity is considered as the macro-value proportional to the concentration of elementary vortex filaments inhabiting the micro-level. The vortex motion of a viscous medium is represented as the stochastic motion of an infinite set of elementary vortex filaments on the micro-level governed by the stochastic differential equations, where the stochastic velocity component of every filament simulates the viscous diffusion of vorticity, and the regular component is the macro-value induced according to the Biot–Savart law and simulates the convective transfer of vorticity.

In flows with boundaries, the production of elementary vortex filaments at the boundary is introduced to satisfy the no-slip condition. This phenomenon is described by the application of the generalized Markov processes theory. The integral equation for the production intensity of elementary vortex filaments is derived and solved using the no-slip condition reformulated in terms of vorticity. Additional conditions on this intensity are determined to avoid the many-valuedness of the pressure in a multi-connected flow domain. This intensity depends on the vorticity in the flow and the boundary velocity at every time instant, together with boundary acceleration.

As a result, the successive and accurate application of the product-integral method allows the study of vortex dynamics in a viscous fluid according to the concepts of Helmholtz and Kelvin.

A row of two-dimensional vortices forms in an offshore zone when regular surface waves run up a sloping flat bed. This vortex row is called the offshore vortex train. The vortices begin to appear near the breaking point. Moving in the offshore direction, they develop and increase their horizontal lengthscale through vortex merging. After reaching a particular offshore location, however, they decay rapidly. The formation region of the vortex train has been investigated on the basis of visual experiments for three bed slopes. Its formation does not depend on the type of wave breaking but is observed when the steepness of deep-water waves is smaller than 4.2 × 10−2. The horizontal lengthscale of the vortices and the velocities of the vortex movement have also been evaluated empirically.

The familiar problem of the propagation of surface waves over variable depth is reconsidered. The surface wave is taken to be a slowly evolving nonlinear wave (governed by the Korteweg–de Vries equation) and the depth is also assumed to be slowly varying; the fluid is stationary in its undisturbed state. Two cases are addressed: the first is where the scale of the depth variation is longer than that on which the wave evolves, and the second is where it is shorter (but still long). The first case corresponds to that discussed by a number of previous authors, and is the problem which has been approached through the perturbation of the inverse scattering transform method, a route not followed here. Our more direct methods reveal a new element in the solution: a perturbation of the primary wave, initiated by the depth change, which arises at the same order as the left-going shelf. The resulting leading-order mass balance is described, with more detail than hitherto (made possible by the use of a special depth variation). The second case is briefly presented using the same approach, and some important similarities are noted.

An axisymmetric vortex-sheet model is applied to simulate an experiment of Didden (1979) in which a moving piston forces fluid from a circular tube, leading to the formation of a vortex ring. Comparison between simulation and experiment indicates that the model captures the basic features of the ring formation process. The computed results support the experimental finding that the ring trajectory and the circulation shedding rate do not behave as predicted by similarity theory for starting flow past a sharp edge. The factors responsible for the discrepancy between theory and observation are discussed.

The stabilities of salt-finger and plume convection, two major flows characterizing the fluid dynamics of NH4Cl solutions cooling from below, are investigated by theoretical and experimental approaches. A linear stability analysis is implemented to study theoretically the onset of salt-finger convection. Special emphasis is placed on the competition between different instability modes. It is found that in most of the cases considered, the neutral curve consists of two separated monotonic branches with a Hopf bifurcation branch in between; the right-hand monotonic branch corresponding to the boundary-layer-mode convection is more unstable than the left-hand monotonic branch corresponding to the mushy-layer mode. We also conducted a series of experiments covering wide ranges of bulk fluid concentration C∞ and bottom temperature TB to study the stability characteristics of plume convection. From the measurement of both temperature and concentration of the interstitial fluid in the mushy layer, we verify that during the progress of solidification the melt in the mush is in a thermodynamic equilibrium state except at the melt/mush interface where most of the solidification occurs. The critical Rayleigh number of the onset of plume convection is found to be Rccm = 1.1 × 107Π* (see (22)), where Π* is the permeability of the mush. This relation is believed to be valid up to supereutectic NH4Cl solutions.

Two-dimensional Langmuir circulation in a layer of stably stratified water and the mathematically analogous problem of double-diffusive convection are studied with mixed boundary conditions. When the Biot numbers that occur in the mechanical boundary conditions are small and the destabilizing factors are large enough, the system will be unstable to perturbations of large horizontal length. The instability may be either direct or oscillatory depending on the control parameters. Evolution equations are derived here to account for both cases and for the transition between them. These evolution equations are not limited to small disturbances of the nonconvective basic velocity and temperature fields, provided the spatial variations in the horizontal remain small. The direct bifurcation may be supercritical or subcritical, while in the case of oscillatory motions, stable finite-amplitude travelling waves emerge. At the transition, travelling waves, standing waves, and modulated travelling waves all are stable in sub-regimes.

In view of several practical ramifications of this problem, computational-analytical techniques for calculating waves induced by heaving arbitrary bodies in narrow tanks have been developed, including nonlinear wave groups produced near tank resonance. These feature computational near-field solutions matched with appropriate far-field solutions. In the linear case, the far field is provided by linear mode superposition. In the nonlinear case, the far field is described by a suitable nonlinear evolution equation of the cubic Schrödinger type. Matching techniques were developed. Calculations were successfully carried out and the results confirm the important effect of tank walls on added mass and damping.

Results of computations have been compared with some data obtained with a conical wavemaker in a narrow tank. Pronounced nonlinear wave groups were obtained near resonance, and these are well reproduced in some detail by the nonlinear theory and computations, without considering any effects of dissipation.

The related problem of resonant wave groups produced by a segmented paddle wavemaker has also been treated by analysis and subject to computation, with good general agreement with past experiments. The technique features matching near- and far-field computations using energy considerations.

It is well known that the widely used powerful geostrophic equations that single out the vertical component of the Earth's rotation cease to be valid near the equator. Through a vorticity and an angular momentum analysis on the sphere, we show that if the flow varies on a horizontal scale L smaller than (Ha)1/2 (where H is a vertical scale of motion and a the Earth's radius), then equatorial dynamics must include the effect of the horizontal component of the Earth's rotation. In equatorial regions, where the horizontal plane aligns with the Earth's rotation axis, latitudinal variations of planetary angular momentum over such scales become small and approach the magnitude of its radial variations proscribing, therefore, vertical displacements to be freed from rotational constraints. When the zonal flow is strong compared to the meridional one, we show that the zonal component of the vorticity equation becomes (2Ω.Δ)u1 = g/ρ0)(∂ρ/a∂θ). This equation, where θ is latitude, expresses a balance between the buoyancy torque and the twisting of the full Earth's vorticity by the zonal flow u1. This generalization of the mid-latitude thermal wind relation to the equatorial case shows that u1 may be obtained up to a constant by integrating the ‘observed’ density field along the Earth's rotation axis and not along gravity as in common mid-latitude practice. The simplicity of this result valid in the finite-amplitude regime is not shared however by the other velocity components.

Vorticity and momentum equations appropriate to low frequency and predominantly zonal flows are given on the equatorial β-plane. These equatorial results and the mid-latitude geostrophic approximation are shown to stem from an exact generalized relation that relates the variation of dynamic pressure along absolute vortex lines to the buoyancy field. The usual hydrostatic equation follows when the aspect ratio δ = H/L is such that tan θ/δ is much larger than one. Within a boundary-layer region of width (Ha)1/2 and centred at the equator, the analysis shows that the usually neglected Coriolis terms associated with the horizontal component of the Earth's rotation must be kept.

Finally, some solutions of zonally homogeneous steady equatorial inertial jets are presented in which the Earth's vorticity is easily turned upside down by the shear flow and the correct angular momentum ‘Ωr2cos2(θ)+u1rCos(θ)’ contour lines close in the vertical–meridional plane.

The steady axisymmetric flow generated in an unbounded incompressible viscous fluid, of density ρ and kinematic viscosity v, by torque-producing singularities with constant line density c along the semi-infinite line θ = 0 of a spherical polar coordinate system (r, θ, ϕ) that was investigated by Paull & Pillow (1985b), is reconsidered. The numerical solution constructed revealed the following features. (i) For values of c up to about 46.9 there is only one solution where the axial component of the meridional flow is directed from θ = 0 to θ = π. This solution can be continued to all values of c. (ii) For c > 46.9 the system of equations allows bifurcation and two more solutions with a single separatrix are possible. (iii) For c = ∞ one of the two branches of the separatrix asymptotes to θ = ½π and the other to θ = π. The asymptotic solution for large c constructed by Paull & Pillow (1985b), where the meridional flow consists of two colliding flows, relates to the bifurcation solution where the separatrix asymptotes to ½π as c → ∞.

This paper is concerned with the stability of steady inviscid flows with closed streamlines. In increasing order of complexity we look at two-dimensional planar flows, poloidal (r, z) flows, and swirling recirculating flows. In each case we examine the relationship between Arnol’d's variational approach to stability, Moffatt's magnetic relaxation technique, and a more recent relaxation procedure developed by Vallis et al. We start with two-dimensional (x, y) flows. Here we show that Moffatt's relaxation procedure will, under a wide range of circumstances, produce Euler flows which are stable. The physical reasons for this are discussed in the context of the well-known membrane analogy. We also show that there is a close relationship between Hamilton's principle and magnetic relaxation. Next, we examine poloidal flows. Here we find that, by and large, our planar results also hold true for axisymmetric flows. In particular, magnetic relaxation once again provides stable Euler flows. Finally, we consider swirling recirculating flows. It transpires that the introduction of swirl has a profound effect on stability. In particular, the flows produced by magnetic relaxation are no longer stable. Indeed, we show that all swirling recirculating Euler flows are potentially unstable to the extent that they fail to satisfy Arnol’d's stability criterion. This is, perhaps, not surprising, as all swirling recirculating flows include regions where the angular momentum decreases with radius and we would intuitively expect such flows to be prone to a centrifugal instability. The paper concludes with a discussion of marginally unstable modes in swirling flows. In particular, we examine the extent to which Rayleigh's original ideas on stability may be generalized, through the use of the Routhian, to include flows with a non-zero recirculation.

A stagnation point flow of the form U = (0, Ay, — Az) is unstable to three-dimensional disturbances. It has been shown that the vorticity components of such a disurbance that are perpendicular to the direction of the diverging flow will decay, and that the parallel component of vorticity can grow. We augment these findings by showing that fully nonlinear steady-state deviations from this flow exist that consist of a periodic distribution of counter-rotating vortices whose axes lie parallel to the direction of the diverging flow. These solutions have two independent parameters: the dimensionless strength of the converging flow, and the intensity of the vortices. We examine the structure of these vortices in the asymptotic limits of large strain rate of the converging flow, and of large amplitude of the vortices.

In homogeneous and density-stratified inviscid shear flows, the mechanism for instability that is most commonly invoked and discussed is Kelvin–Helmholtz instability, as it occurs for a simple velocity discontinuity. There is a second mechanism, the wave-interaction mechanism, which is much more general, and is the subject of this paper. This mechanism depends on two free waves that propagate in opposite directions in a stratified shear flow, and which may become stationary relative to each other because of the shear. If this occurs, and their relative phase is suitably chosen, the velocity field of each wave increases the displacement of the other, and so the disturbance grows.

We show that this mechanism is responsible for instability in a general class of symmetric but otherwise arbitrary velocity and density profiles, provided that the Richardson number Ri < ¼ in a central region of arbitrarily small thickness. A critical layer exists in this central region for the growing disturbance, but its role in the instability process is incidental. When Ri > ¼ everywhere, the flow is stable because the free waves described above are absorbed by the critical layer, and hence are heavily damped. The necessary criteria of Rayleigh and Fjortoft for instability in homogeneous fluid are seen to provide a suitable geometry for two interacting waves. Some specific examples are given, including a succinct explanation of Holmboe waves.

The effects of crossflow on the growth rate of inviscid Görtler vortices in a hypersonic boundary layer with pressure gradient are studied in this paper. Attention is focused on the inviscid mode trapped in the temperature adjustment layer; this mode has greater growth rate than any other mode at the minimum order of the Görtler number at which Görtler vortices may exist. The eigenvalue problem which governs the relationship between the growth rate, the crossflow amplitude and the wavenumber is solved numerically, and the results are then used to clarify the effects of crossflow on the growth rate of inviscid Görtler vortices. It is shown that crossflow effects stabilize Görtler vortices in different manners for incompressible and hypersonic flows. The neutral mode eigenvalue problem is found to have an exact solution, and as a byproduct, we have also found the exact solution to a neutral mode eigenvalue problem which was formulated, but unsolved before, by Bassom & Hall (1991).

This paper treats a surface-tension-driven liquid-metal flow in a cylinder with a steady externally applied non-uniform axisymmetric magnetic field. The top boundary consists of an annular free surface around a solid disk, modelling the Czochralski growth of silicon crystals. A radial temperature gradient produces a decrease of the surface tension from the disk edge to the vertical cylinder wall. The magnetic flux density is sufficiently large that inertial effects and convective heat transfer are negligible. First we present large-Hartmann-number asymptotic solutions for magnetic fields with either a non-zero or a zero axial component at the free surface. The asymptotic solutions indicate that a purely radial magnetic field at the free surface represents a singular limit of more general magnetic fields. Secondly we present numerical solutions for arbitrary values of the Hartmann number, and we treat the evolution of the thermocapillary convection as the axial magnetic field at the free surface is changed continuously from the full field strength to zero.

Experiments on bifurcation of rotating liquid drops into two-lobed shapes were conducted during a Space shuttle flight. The drops were levitated in air and spinned using acoustic fields in the low-gravity environment. These experiments have successfully resolved the discrepancies existing between the previous experimental results and the theoretical predictions. In the simplest case of a rotating drop that is free from deformation by external forces, the results agree well with the existing theoretical predictions. In the case of a rotating drop subjected to flattening by the acoustic radiation stress, deliberately or otherwise, the experiments suggest the existence of a family of curves, with the free drop as the limiting case.