We describe an instability that appears at the front of laminar gravity currents as they intrude into a viscous, miscible ambient fluid. The instability causes a current to segment into fingers aligned with its direction of flow. In the case of currents flowing along a rigid floor into a less dense fluid, the case of primary interest here, two mechanisms can produce this instability. The first is gravitational and arises because the nose of the gravity current is elevated above the floor and overrides a buoyantly unstable layer of ambient liquid. The second is a form of viscous fingering analogous to a Saffman–Taylor instability in a Hele-Shaw cell. Whereas the ambient fluid must be more viscous than the current in order for the latter instability to occur, the gravitational instability can occur even if the ambient fluid is less viscous, as long as it is viscous enough to elevate the nose of the current and trap a layer of ambient fluid. For the gravitational mechanism, which is most important when the current and ambient fluids have comparable viscosities, the wavelength when the instability first appears is proportional to a length scale constructed with the viscosity, the flux and the buoyancy. The Saffman–Taylor-type mechanism is most important when the ambient liquid is much more viscous than the current. We have carried out experiments with miscible fluids in a Hele-Shaw cell that show that, at the onset of instability, the ratio of the finger wavelength to the cell width is a constant approximately equal to 2. This result is explained by using the principle that the flow tends to minimize the dissipation associated with the finger perturbation. For the gravity currents with high viscosity ratios, the ratio of the wavelength to the current thickness is also a constant of about 2, apparently consistent with the same mechanism. But, further analysis of this instability mechanism is required in order to assess its role in wavelength selection for gravity currents.

A series of laboratory experiments was conducted to study the flows and exchange processes generated by turbulent convection in a shallow fluid with a combination of a shelf and slope topography in the presence of rotation. For convenience, heat loss at the ocean surface was modelled by heating from below with a buoyancy flux B0 applied to a circular portion (of radius R) of the base of a cylindrical tank, rotating with angular frequency f. The working volume was closed by an inverted model of a shelf and slope topography (with slope angle ϕ), creating a fluid height H between the forced surface and the shelf. After the initiation of the buoyancy forcing, the average temperature in the actively convecting region initially increases linearly with time but slows down once a lateral heat flux is generated by baroclinic instability at the edge of the convecting region. The wavelength of this instability is described by λ=(5.9±0.3) RD, with RD the Rossby radius of deformation, defined by (g′H)1/2/f, where g′ is the reduced gravity based on the density difference between the convecting and ambient fluids. A steady state is eventually reached when the lateral heat flux balances the (vertical) heat flux due to the forcing. The results differ from previous work in either unbounded or in constant-depth environments. It is shown that the steady-state density anomaly between the convecting and ambient regions is given by g′f=(1.6±0.2) (B0f)1/2 (R/H), while the time to reach this steady state is τ=(3.1±0.5) (f/B0)1/2R. The eddy velocity, characterizing the lateral exchange process, is given by vflux≈1.2 (B0/f)1/2. These results are consistent with the description of the lateral exchange process by eddy diffusion (rather than advection). Comparisons are made between the experimental results and field observations of convection events.

Experiments have been undertaken to characterize the flow field over a delta wing, with an 85° sweep angle, at 12.5° incidence. Application of a laser Doppler anemometer has enabled detailed three-dimensional velocity data to be obtained within the free shear layer, revealing a system of steady co-rotating vortical structures. These sub-vortex structures are associated with low-momentum flow pockets in the separated vortex flow. The structures are found to be dependent on local Reynolds number, and undergo transition to turbulence. The structural features disappear as the sub-vortices are wrapped into the main vortex core. A local three-dimensional Kelvin–Helmholtz-type instability is suggested for the formation of these vortical structures in the free shear layer. This instability has parallels with the cross-flow instability that occurs in three-dimensional boundary layers. Velocity data at high Reynolds numbers have shown that the sub-vortical structures continue to form, consistent with flow visualization results over fighter aircraft at flight Reynolds numbers.

The resonant scattering of topographically trapped, low-mode progressive edge waves by longshore periodic topography is investigated using a multiple-scale expansion of the linear shallow water equations. Coupled evolution equations for the slowly varying amplitudes of incident and scattered edge waves are derived for small-amplitude, periodic depth perturbations superposed on a plane beach. In ‘single-wave scattering’, an incident edge wave is resonantly scattered into a single additional progressive edge wave having the same or different mode number (i.e. longshore wavenumber), and propagating in the same or opposite direction (forward and backward scattering, respectively), as the incident edge wave. Backscattering into the same mode number as the incident edge wave, the analogue of Bragg scattering of surface waves, is a special case. In ‘multi-wave scattering’, simultaneous forward and backward resonant scattering results in several (rather than only one) new progressive edge waves. Analytic solutions are obtained for single-wave scattering and for a special case of multi-wave scattering involving mode-0 and mode-1 edge waves, over perturbed depth regions of both finite and semi-infinite longshore extent. In single-wave backscattering with small (subcritical) detuning (i.e. departure from exact resonance), the incident and backscattered wave amplitudes both decay exponentially with propagation distance over the periodic bathymetry, whereas with large (supercritical) detuning the amplitudes oscillate with distance. In single-wave forward scattering, the wave amplitudes are oscillatory regardless of the magnitude of the detuning. Multi-wave solutions combine aspects of single-wave backward and forward scattering. In both single- and multi-wave scattering, the exponential decay rates and oscillatory wavenumbers of the edge wave amplitudes depend on the detuning. The results suggest that naturally occurring rhythmic features such as beach cusps and crescentic bars are sometimes of large enough amplitude to scatter a significant amount of incident low-mode edge wave energy in a relatively short distance (O(10) topographic wavelengths).

This paper presents a numerical study of the dynamics and stability of two-dimensional thermal plumes in a significantly stratified layer. Motivated by stellar envelope convection in which radiative cooling at the star's photosphere drives vigorous down flows, we examine cool plumes descending through an adiabatically stratified layer of increasing density with depth. Such flows are inaccessible by laboratory experiments, yet are important to the understanding of heat and momentum transport, magnetic field generation, and acoustic excitation in stars like the Sun. We find that the structure of thermal plumes in a stratified compressible medium is significantly different from that in an incompressible one, with pressure perturbations playing an important dynamical role. Additionally, we find that the plumes are subject to vigorous secondary instabilities even in a quiescent background medium. While the flows studied are not fully turbulent but transitional, the nature of the compressive instabilities and their influence on subsequent flow evolution suggests that advective detrainment of fluid from the plume region results. Simplified plume models assuming a hydrostatic pressure distribution and velocity-proportional entrainment may thus be inappropriate in this context.

The motion of membrane-bound objects is important in many aspects of biology and physical chemistry. A hydrodynamic model for this Fconfiguration was proposed by Saffman & Delbrück (1975) and here it is extended to study the translation of a disk-shaped object in a viscous surface film overlying a fluid of finite depth H. A solution to the flow problem is obtained in the form of a system of dual integral equations that are solved numerically. Results for the friction coefficient of the object are given for a complete range of the two dimensionless parameters that describe the system: the ratio of the sublayer (η) to membrane (ηm) viscosities, Λ=ηR/ηmh (where R and h are the object radius and thickness of the surface film, respectively), and the sublayer thickness ratio, H/R. Scaling arguments are presented that predict the variation of the friction coefficient based upon a comparison of the different length scales that appear in the problem: the geometric length scales H and R, the naturally occurring length scale [lscr ]m=ηmh/η, and an intermediate length scale [lscr ]H= (ηmhH/η)1/2. Eight distinct asymptotic regimes are identified based upon the different possible orderings of these length scales for each of the two limits Λ[Lt ]1 and Λ[Gt ]1. Moreover, the domains of validity of available approximations are established. Finally, some representative surface velocity fields are given and the implication of these results for the characterization of hydrodynamic interactions among membrane-bound proteins adjacent to a finite-depth sublayer is discussed briefly.

The steady undertow created by waves breaking at a beach and slowly flowing offshore can become unstable and create a train of submerged offshore migrating vortices with shorter length scales and longer time scales than the incident waves, as shown by Matsunaga, Takehara & Awaya (1988, 1994). These vortices rotate about horizontal axes parallel to the shoreline. Our larger-scale laboratory experiments show that an additional layer of vortices can exist over the water depth, with vortices near the water surface rotating in the same direction as the wave-induced water particle trajectories, while those located at about mid-depth rotate in the opposite direction.

A theoretical and numerical analysis shows that these vortices are due to instabilities of the undertow. Far offshore of the surf zone, the vortex trains decay because the velocity profile for the undertow becomes linear over depth, hence neutrally stable to any disturbances.

The effects of the intensity and frequency of a time-periodic external field on the rotary motion of a dipolar spherical particle suspended in homogeneous shear are studied with the goal of providing insight into problems concerning the motion of swimming microorganisms and the macroscopic behaviour of ferrofluids. The analysis reveals two modes of motion: convergence of the particle to a global time-periodic attractor, and quasi-periodic motion. The former mode of particle rotation generally appears for sufficiently strong fields. However, asymptotic analysis clarifies that it may occur even for very weak fields as a cumulative result of appropriate resonance interactions.

A sufficient condition for the occurrence of a global time-periodic attractor is established for an external field acting in the plane of shear. Asymptotic results together with numerical evidence indicate that this condition is in fact a necessary condition as well. Making use of this condition we obtain the division of the plane of parameters into domains respectively corresponding to quasi-periodic motion and global time-periodic attractors. The latter domain has the structure of non-intersecting Arnold's tongues. Throughout each, the average frequency of dipole rotation about the vorticity vector is a constant (integral) multiple of the forcing frequency (frequency locking). In the case of quasi-periodic motion, there simultaneously coexist separate domains in orientation space where the rotary motion is locally characterized by different constant rotation numbers. These may assume both rational and irrational values. Potential implications of the distinction between these modes of rotary motion on the characterization of effective (macroscale) ferrofluid properties are briefly discussed.

A study is made of the nonlinear spatial evolution of an externally excited instability wave in a mixing layer of nearly perfectly conducting fluid with a large Reynolds number in a weak parallel magnetic field.

It is shown that the evolution pattern bears a resemblance to that of disturbances in a weakly stratified shear flow with the Prandtl number less than unity which was studied in our earlier publication (Shukhman & Churilov 1997): a weak magnetic field, like a weak stratification when Pr<1, has a stabilizing effect on the nonlinear development of disturbances and in the case when the linear growth rate of the wave is not too large leads either to the instability saturation in the viscous critical layer regime or to the establishment of a unsteady nonlinear critical layer regime where the wave amplitude oscillates without exceeding a certain maximum value. In this case the regime of the quasi-steady nonlinear critical layer is not attained evolutionarily. When the linear growth rate is large enough the magnetic field has no dynamical effect on evolution and the quasi-steady nonlinear critical layer regime with the well-known power-law growth of amplitude (A∝x2/3) is eventually attained.

Also, the critical layer structure and the evolution behaviour in the case of a strong difference of dissipation coefficients (i.e. ordinary viscosity and magnetic viscosity) are considered.

We calculate spatially and temporally periodic standing waves using a spectral boundary integral method combined with Newton iteration. When surface tension is neglected, the non-monotonic behaviour of global wave properties agrees with previous computations by Mercer & Roberts (1992). New accurate results near the limiting form of gravity waves are obtained by using a non-uniform node distribution. It is shown that the crest angle is smaller than 90° at the largest calculated crest curvature. When a small amount of surface tension is included, the crest form is changed significantly. It is necessary to include surface tension to numerically reproduce the steep standing waves in Taylor's (1953) experiments. Faraday-wave experiments in a large-aspect-ratio rectangular container agree with our computations. This is the first time such high-amplitude, periodic waves appear to have been observed in laboratory conditions. Ripple formation and temporal symmetry breaking in the experiments are discussed.

We examine the dynamics of two-dimensional steep and breaking standing waves generated by Faraday-wave resonance. Jiang et al. (1996) found a steep wave with a double-peaked crest in experiments and a sharp-crested steep wave in computations. Both waveforms are strongly asymmetric in time and feature large superharmonics. We show experimentally that increasing the forcing amplitude further leads to breaking waves in three recurrent modes (period tripling): sharp crest with breaking, dimpled or flat crest with breaking, and round crest without breaking. Interesting steep waveforms and period-tripled breaking are related directly to the nonlinear interaction between the fundamental mode and the second temporal harmonic. Unfortunately, these higher-amplitude phenomena cannot be numerically modelled since the computations fail for breaking or nearly breaking waves. Based on the periodicity of Faraday waves, we directly estimate the dissipation due to wave breaking by integrating the support force as a function of the container displacement. We find that the breaking events (spray, air entrainment, and plunging) approximately double the wave dissipation.

In this paper we consider the flow field within and around a vortex as it ‘collides’ with a thin plate at a right angle to its axis of rotation. We show that based solely on inviscid flow theory, vorticity in the core of the vortex is redistributed significantly. The main cause of this redistribution is the presence of axial flow within the vortex; we call this vortical structure which contains axial flow a vortex–jet. In this work we show that when the axial velocity within the vortex is toward the plate, vorticity is redistributed radially outward from the core resulting in a significant reduction in the axial vorticity there; the vortex is said to ‘bulge’ reflecting an increase in the nominal vortex core radius. A by-product of this interaction is that the suction peak amplitude caused by the presence of the vortex rapidly decreases and the pressure soon returns to a quasi-steady distribution. On the other hand, when the axial velocity within the vortex is directed away from the surface, the suction peak persists and the vortex core radius decreases. The numerical results were validated by comparison with an analytical solution for a sinusoidal vortex jet. Analytical solutions were also derived for the initial and final states of a pure jet; the numerical results are strongly supported by the analysis. In addition, all of these results are consistent with experiments, and their relevance to the interaction between a tip vortex and a helicopter airframe is also discussed.

We extend Floquet's Theorem, similar to that used in calculating electronic and optical band gaps in solid state physics (Bloch's Theorem), to derive dispersion relations for small-amplitude water wave propagation in the presence of an infinite array of periodically arranged surface scatterers. For one-dimensional periodicity (stripes), we find band gaps for wavevectors in the direction of periodicity corresponding to frequency ranges which support only non-propagating standing waves, as a consequence of multiple Bragg scattering. The dependence of these gaps on scatterer strength, density, and water depth is analysed. In contrast to band gap behaviour in electronic, photonic, and acoustic systems, we find that the gaps here can increase with excitation frequency ω. Thus, higher-order Bragg scattering can play an important role in suppressing wave propagation. In simple two-dimensional periodic geometries no complete band gaps are found, implying that there are always certain directions which support propagating waves. Evanescent modes offer one qualitative reason for this finding.

The head-one interaction of a supersonic streamwise vortex with a circular cylinder reveals a vortex breakdown similar in many ways to that of incompressible vortex breakdown. In particular, the dramatic flow reorganization observed during the interaction resembles the conical vortex breakdown reported by Sarpkaya (1995) at high Reynolds number. In the present study, vortex breakdown is brought about when moderate and strong streamwise vortices encounter the bow shock in front of a circular cylinder at Mach 2.49. The main features of the vortex/cylinder interaction are the formation of a blunt-nosed conical shock with apex far upstream of the undisturbed shock stand-off distance, and a vortex core which responds to passage through the apex of the conical shock by expanding into a turbulent conical flow structure. The geometry of the expanding vortex core as well as the location of the conical shock apex are seen to be strong functions of the incoming vortex strength and the cylinder diameter. A salient feature of the supersonic vortex breakdown is the formation of an entropy-shear layer, which separates an interior subsonic zone containing the burst vortex from the surrounding supersonic flow. In keeping with the well-established characteristics of the low-speed vortex breakdown, a region of reversed flow is observed inside the turbulent subsonic zone. The steady vortex/cylinder interaction flow fields generated in the current study exhibit many characteristics of the unsteady vortex distortion patterns previously observed during normal shock wave/vortex interactions. This similarity of the instantaneous flow structure indicates that the phenomenon previously called vortex distortion by Kalkhoran et al. (1996) is a form of supersonic vortex breakdown.

Journal of Fluid Mechanics, vol. 321 (1996), pp. 1–24