The main purpose of this paper is to provide some carefully documented experimental observations of the subharmonic generation of edge waves over a plane beach by waves normally incident on the beach from the distant ocean. In order to establish experimentally the details of the subharmonic instability mechanism, it is important first to determine the properties of the primary wave field whose stability is to be investigated. Thus, a detailed appraisal has been made of the wave field established in the tank and over the beach in the absence of edge waves. These data have been particularly useful in defining the amplitudes of the incident- and reflected-wave fields at the toe of the beach, the magnitudes of which are central to the triggering of the edge-wave instability. Details are presented of the edge-wave field over the beach, as well as marginal stability curves and initial growth rates of the instability, the latter two of which are compared with theoretical estimates obtained from extant theories of the instability mechanism. Some experiments are also described in which edge waves were established as modes forced by topographical imperfections at the beach.

Some of the effects of a passive, single-layer, viscoelastic compliant surface on the stability of a Blasius boundary layer were investigated in a low-turbulence wind tunnel. Measurements of the wavelength and growth rates of vibrating-ribbon-excited harmonic waves were made by hot-wire anemometry. The data for three compliant surfaces with different shear moduli, material damping coefficients, and thicknesses were compared to rigid-surface data. The flow-induced surface displacements were measured using an electro-optic displacement transducer. The results show that the growth rates of unstable Tollmien–Schlichting waves, and the extent of the unstable region in the (F, Rδ*)-plane are reduced over the compliant surfaces relative to those over a rigid surface with the absence of flow-induced surface instabilities. The suppression of the Tollmien–Schlichting waves is accompanied by a surface motion driven by the flow field at the excitation frequency. The experimental results suggest that a delay of the onset to turbulence is possible in air by using appropriately tuned surface characteristics. Further experiments are needed to study the three-dimensional disturbance mode, the flow-induced surface instabilities and the breakdown process.

The neutral stability curve for the flat-plate boundary layer has been calculated using the Orr–Sommerfeld equation and compared to those obtained using upper- and lower-branch scalings. The Orr–Sommerfeld results agree well with the lower-branch scaling at Reynolds numbers relevant to experiment, but agree well with the upper-branch scaling only for Rδ > 105. It is shown that the critical layer only emerges from the viscous wall layer when Rδ > 105. This suggests that for Rδ < 105, when the critical layer lies within the viscous wall layer, the disturbance has a triple-deck structure, even for the upper branch of the neutral curve (which can be reached if the phase jump across the critical layer is retained).

The transition from a triple-deck to a five-deck structure with increasing Reynolds number on the upper branch occurs relatively abruptly and can be associated with a square-root branch point in the Tietjens function. Essentially, the lower- and upper-branch scalings pertain to two different modes, the first possessing a triple-deck structure, the second a five-deck structure. The modes are connected at the branch point, and the neutral curves of each mode join to give a single curve close to this branch point. The asymptotic expansions for the upper- and lower-branch neutral curves depend upon the analyticity of the dispersion relationship, and so the proximity of the branch point indicates where these expansions will be liable to inaccuracies. This explains the poor neutral-curve predictions made by five-deck analyses at the Reynolds numbers where transition occurs.

The fingering instabilities in vertical miscible displacement flows in porous media driven by both viscosity and density contrasts are studied using linear stability analysis and direct numerical simulations. The conditions under which vertical flows are different from horizontal flows are derived. A linear stability analysis of a sharp interface gives an expression for the critical velocity that determines the stability of the flow. It is shown that the critical velocity does not remain constant but changes as the two fluids disperse into each other. In a diffused profile, the flow can develop a potentially stable region followed downstream by a potentially unstable region or vice versa depending on the flow velocity, viscosity and density profiles, leading to the potential for ‘reverse’ fingering. As the flow evolves into the nonlinear regime, the strength and location of the stable region changes, which adds to the complexity and richness of finger propagation. The flow is numerically simulated using a Hartley-transform-based spectral method to study the nonlinear evolution of the instabilities. The simulations are validated by comparing to experiments. Miscible displacements with linear density and exponential viscosity dependencies on concentration are simulated to study the effects of stable zones on finger propagation. The growth rates of the mixing zone are parametrically obtained for various injection velocities and viscosity ratios.

The near-contact axisymmetric electrophoretic motion of a pair of spherical particles with thin electric double layers and differing surface zeta-potentials is analysed for low Reynolds numbers and moderate surface potentials. Near-contact electrophoretic motion of a spherical particle normal to a planar conducting boundary is analysed under the same assumptions. Pairwise motion is computed by considering touching particles in point contact; relative motion is described by a perturbation about the touching state using lubrication theory. Analytical formulae are derived for two particles of disparate sizes, and for the motion of a single particle towards a boundary; numerical calculations are performed for all size ratios. The results have a universal form with respect to the particle zeta-potentials. All results indicate that the electrophoresis is a much more efficient mechanism of near-contact motion than is buoyancy. An explanation for this finding is given in terms of the electro-osmotic slip velocity on the particle surfaces that facilitates fluid removal from between approaching surfaces.

The unsteady, three-dimensional, incompressible, viscous flow interactions between a vortical (initially cylindrical) structure advected by a uniform free stream and a spherical particle held fixed in space is investigated numerically for a range of particle Reynolds numbers 20 ≤ Re ≤ 100. The counter-clockwise rotating vortex tube is initially located ten sphere radii upstream from the sphere centre. The finite-difference computations yield the flow properties and the temporal distributions of lift, drag, and moment coefficients of the sphere. Initially, the lift force is positive owing to the upwash on the sphere, then becomes negative owing to the downwash as the vortex tube passes the sphere. Varying the size of the vortex core (σ) shows that the r.m.s. lift coefficient is linearly proportional to the circulation of the vortex tube at small values of σ. At large values of σ, the r.m.s. lift coefficient is linearly proportional to the maximum fluctuation velocity (vmax) induced by the vortex tube but independent of σ. For intermediate values of σ, the r.m.s. lift coefficient depends on both σ and vmax (or equivalently both σ and the circulation). We observe some interesting flow phenomena in the near wake as a function of time owing to the passage of the vortex tube.

In this paper the dynamics of geostrophic flows localized in a thin layer of continuously stratified fluid, which overrides a thick homogeneous layer are studied. The displacement of isopycnal surfaces is assumed large; the β-effect is strong, i.e. \[(R_0/R_e)\cot\theta\gtrsim\epsilon, \] where ε is the Rossby number, θ is the latitude; Re is the Earth's radius, and R0 is the deformation radius based on the total depth of the ocean. An asymptotic system of equations is derived and used to study the stability of zonal currents. Three sufficient conditions of stability are obtained, which restrict the slope of the interface between the stratified and non-stratified layers. The results obtained are applied to the subtropical and subarctic frontal currents in the Northern Pacific: the former was found to be stable, the latter was found to be unstable. However, the growth rate of the instability is very small (the effective time of growth is about 2 years).

This paper examines the baroclinic instability of a quasi-geostrophic flow with vertical shear in a continuously stratified fluid. The flow and density stratification are both localized in a thin upper layer. (i) Disturbances whose wavelength is much smaller than the deformation radius (based on the depth of the upper layer) are demonstrated to satisfy an ‘equivalent two-layer model’ with properly chosen parameters. (ii) For disturbances whose wavelength is of the order of, or greater than, the deformation radius we derive a sufficient stability criterion. The above analysis is applied to the subtropical and subarctic frontal currents in the Northern Pacific. The effective time of growth of disturbances (i) is found to be 16–22 days, the characteristic spatial scale is 130–150 km.

New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier–Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments.

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.

As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.

The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.

An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > Rc [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.

Recent studies have demonstrated the strong influence of end effects on low-Reynoldsnumber bluff body wakes, and a number of questions remain concerning the intrinsic nature of three-dimensional phenomena in two-dimensional configurations. Some of them are answered by the present study which investigates the wake of bluff rings (i.e. bodies without ends) both experimentally and by application of the phenomenological Ginzburg–Landau model. The model turns out to be very accurate in describing qualitative and quantitative observations in a large Reynolds number interval. The experimental study of the periodic vortex shedding regime shows the existence of discrete shedding modes, in which the wake takes the form of parallel vortex rings or ‘oblique’ helical vortices, depending on initial conditions. The Strouhal number is found to decrease with growing body curvature, and a global expression for the Strouhal–Reynolds number relation, including curvature and shedding angle, is proposed, which is consistent with previous straight cylinder results. A secondary instability of the helical modes at low Reynolds numbers is discovered, and a detailed comparison with the Ginzburg–Landau model identifies it as the Eckhaus modulational instability of the spanwise structure of the near-wake formation region. It is independent of curvature and its clear observation in straight cylinder wakes is inhibited by end effects.

The dynamical model is extended to higher Reynolds numbers by introducing variable parameters. In this way the instability of periodic vortex shedding which marks the beginning of the transition range is characterized as the Benjamin–Feir instability of the coupled oscillation of the near wake. It is independent of the shear layer transition to turbulence, which is known to occur at higher Reynolds numbers. The unusual shape of the Strouhal curve in this flow regime, including the discontinuity at the transition point, is qualitatively reproduced by the Ginzburg–Landau model. End effects in finite cylinder wakes are found to cause important changes in the transition behaviour also: they create a second Strouhal discontinuity, which is not observed in the present ring wake experiments.

The linearized inertial instability of the parallel shear flow of a viscoelastic liquid is considered. An elastic Rayleigh equation is derived, for high Reynolds numbers and high Weissenberg numbers, and for a viscoelastic liquid whose first normal stress dominates other stresses. The equation is used to investigate the stability of a submerged jet, that may be planar or axisymmetric, having a parabolic velocity profile. The sinuous mode is found to be fully stabilized by sufficiently large elasticity. The varicose mode in the planar case is partially stabilized, being unstable only at longer wavelengths and with a reduced growth rate. An axisymmetric jet, which is stable to varicose perturbations at zero elasticity, is found to be unstable to short-wave disturbances for small non-zero elasticity. This novel instability involves elastic waves in the shear. It is also present in other modes but does not have the fastest growth rate.

We measure the threshold accelerations necessary to excite surface waves in a vertically vibrated fluid container (the Faraday instability). Under the proper conditions, the thresholds and onset wavelengths agree with recent theoretical predictions for a laterally infinite, finite-depth container filled with a viscous fluid. Experimentally, we show that by using a viscous, non-polar fluid, the finite-size effects of sidewalls and the effects of surface contamination can be made negligible. We also show that finite-size corrections are of order h/L, where h is the fluid depth and L the container size. Based on these measurements, one can more easily interpret certain unexpected observations from previous experimental studies of the Faraday instability.

The modified Boussinesq equations given by Nwogu (1993a) are rederived in terms of a velocity potential on an arbitrary elevation and the free surface displacement. The optimal elevation where the velocity potential should be evaluated is determined by comparing the dispersion and shoaling properties of the linearized modified Boussinesq equations with those given by the linear Stokes theory over a range of depths from zero to one half of the equivalent deep-water wavelength. For regular waves consisting of a finite number of harmonics and propagating over a slowly varying topography, the governing equations for velocity potentials of each harmonic are a set of weakly nonlinear coupled fourth-order elliptic equations with variable coefficients. The parabolic approximation is applied to these coupled fourth-order elliptic equations for the first time. A small-angle parabolic model is developed for waves propagating primarily in a dominant direction. The pseudospectral Fourier method is employed to derive an angular-spectrum parabolic model for multi-directional wave propagation. The small-angle model is examined by comparing numerical results with Whalin's (1971) experimental data. The angular-spectrum model is tested by comparing numerical results with the refraction theory of cnoidal waves (Skovgaard & Petersen 1977) and is used to study the effect of the directed wave angle on the oblique interaction of two identical cnoidal wavetrains in shallow water.

In this paper, the propagation of interfacial waves in a two-layered fluid system is investigated. The interfacial waves are weakly nonlinear and dispersive and propagate in a slowly rotating channel with varying topography and sidewalls, and a weak steady background current field. An evolution equation for the interfacial displacement is derived for waves propagating predominantly in the longitudinal direction of the channel. This new evolution equation is called the unified Kadomtsev–Petviashvili (uKP) equation because most of the KP-type equations existing in the literature for both surface water waves and interfacial waves are special cases of the new evolution equation. The Painlevé PDE test is used to find the conditions under which the uKP equation can be solved by the inverse scattering transform. When these conditions are satisfied, elementary transformations are found to reduce the uKP equation to one of the completely integrable equations: the KP, the Korteweg–de Vries (KdV) or the cylindrical KdV equations. The integral invariants associated with the uKP equation for waves propagating in a varying channel are obtained and their relations with the conservation of mass and energy are discussed.