A microburst can be modelled by releasing a volume of fluid that is slightly heavier than the ambient fluid, allowing it to fall onto a horizontal surface. Vorticity develops on the sides of this parcel as it descends and causes it to roll up into a turbulent vortex ring which impinges on the ground. Such a model exhibits many of the features of naturally occurring microbursts which are a hazard to aviation. In this paper this model is achieved experimentally by releasing a volume of salt water into fresh water from a cylindrical dispenser. When care is taken with the release the spreading rate of the surface outflow is measurable and quite repeatable despite the fact that the flow is turbulent. An elementary numerical approximation to this model, based on inviscid vortex dynamics, has also been developed. A scaling law is proposed which allows experiments with different fluid densities to be compared with each other and with the numerical results. More importantly the scaling law allows us to compare the model results with real microbursts.

The numerical study presented in this work describes the transition from a steady, laminar regime to a periodic regime in an air-filled, two-dimensional cavity subjected to localized heating. The cavity lies at the bottom of a horizontal channel through which a cold air stream flows. The heating is provided by a constant heat input source, located on one of the vertical walls of the cavity and this generates a buoyancy-driven recirculating flow in the cavity. The interaction of this recirculating flow with the cold through-flow leads to steady flow and thermal fields as long as the value of the Grashof number, associated with the heat input from the source, remains below a certain critical value. As this value is exceeded, an unstable situation arises and, after an initial transient, the results show a very regular, periodic, almost sinusoidal behaviour. Similar previous works on natural convective flows in cavities have discovered such a periodic behaviour, as a form of travelling wave instability, having the characteristics of a Hopf bifurcation. These characteristics were also found to be present in this mixed convective situation, where the amplitudes of the oscillating quantities, like the thermal energy outflow, vary with the square root of the Grashof number. However, the present problem is also governed by the value of the Reynolds number, and its effect on the results is studied. The observed instability is of thermal origin, with small fluid cells at the centre of the cavity being subjected to periodic cooling and heating and, therefore, to a circulatory motion due to the density change.

An array of long, vertically uniform salt fingers in an environment with salt input from above, fresh input from below, a vertically constant, stabilizing temperature gradient and negligible salt diffusion is found to be unstable to perturbations with vertical structure. The maximum growth rate and the form of the instability are derived for fingers with widths that yield maximum buoyancy flux in the unperturbed state. The dependence of the instability on the magnitude of the imposed salt difference is obtained for the heat–salt system. A direct (non-oscillatory) mode with a vertical scale of the order of the buoyancy-layer thickness is the most unstable when the amplitude of the vertical velocity of the fingers is large. The instability is due to the shear flow between rising and sinking fluid in adjacent fingers and is relatively unaffected by the perturbation buoyancy. When the driving is weaker, the dominant instability involves the same processes as for the basic fingers, i.e. perturbation buoyancy, viscosity and diffusion, and the mode becomes oscillatory in time. All of the most unstable modes derived here have a vertical scale of the order of the buoyancy-layer thickness. Both the direct and the oscillatory modes have net horizontal flows that vary with the vertical coordinate and time and in finite amplitude could cause the fingers to incline toward the horizontal. The oscillatory mode involves pairs of fingers so the emerging behaviour could include a kind of period doubling.

This paper examines the streamwise dispersion of passive contaminant molecules released in a time-dependent laminar flow through a tube in the presence of boundary absorption or a catalytic wall reaction, which causes a depletion of contaminant in the flow. A finite-difference implicit scheme has been used to solve the unsteady convective–diffusion equation for all time. Here it is shown how the mixing of the cross-sectionally integrated concentration of contaminant molecules is influenced by the frequency of pressure pulsation and the heterogeneous reaction at the boundary. The behaviour of the dispersion coefficient due to the shear effects of steady, oscillatory, and the combined action of steady and periodic currents have been examined separately. The comparison reveals that for all cases the dispersion coefficient asymptotically reaches a stationary state after a certain time and it decreases with the absorption parameter. The increased wall absorption causes negatively skewed deviations from Gaussianity.

Just as mean motions, usually described as acoustic streaming, can be generated by sound waves, so also those cochlear travelling waves into which incident sound waves are converted in the liquid-filled mammalian inner ear are capable of generating mean motions. These predominate, for acoustic components of each frequency ω, near the characteristic place where the wave energy E per unit length rises rather steeply to a maximum Emax before dropping precipitously to zero.

Even though the nature of cochlear travelling waves, as determined (above all) by the sharply and continuously falling distribution of stiffness for the basilar membrane vibrating within the cochlear fluids, is very different from that of ordinary sound waves (see §§2, 3 and 4 respectively for energy distribution along the length of the cochlea, over a cochlear cross-section and within boundary layers), nevertheless a comprehensive analysis of mean streaming motions in the cochlea shows them to be governed by remarkably similar laws. The expression \[ {\textstyle\frac{1}{4}}V^2c^{-1}-{\textstyle\frac{3}{4}}V({\rm d}V/{\rm d}x)\,\omega^{-1} \] (equation (1)) appropriate to a wave travelling in the x-direction with velocity amplitude V(x), as obtained by Rayleigh (1896) for the mean acoustic-streaming velocity just outside a boundary layer due to wave dissipation therein, remains a good approximation (see §§5 and 6 – with some modest corrections, at low or at high wavenumbers respectively, analysed in §§7 and 8) for travelling waves in the cochlea; where, however, the decrease of their phase velocity c to low values near the characteristic place conspires with the increase of V to enhance streaming there.

Farther from the boundary layer attached to the basilar membrane, the mean streaming is derived (§9) as a low-Reynolds-number motion compatible with the distribution (1) of ‘effective slip velocity’ at the boundary. This velocity's precipitous fall to zero at the characteristic place is shown (§§9 and 10) to produce there a mean volume outflow \[ q = \frac{0.15 E^{\max}}{\rho(\omega\nu)^{\frac{1}{2}}L} \] (equation (160)) per unit length of the basilar membrane into the scala media; here, ρ and ν are the endolymph's density and kinematic viscosity (essentially, those of water) and L is the e-folding distance for basilar-membrane stiffness.

Equation (160), derived here for a freely propagating wave (and so not allowing for enhancements from any travelling-wave amplification – discussed qualitatively in §3 – due to forcing by vibrations of outer hair cells) is the main conclusion of this paper. Physiological questions of whether this flow q may be channelled through the space between the tectorial membrane and inner hair cells, whose stereocilia may therefore be stimulated by a mean deflecting force, are noted here but postponed for detailed consideration in a later paper.

Laboratory experiments reveal that, for increasing time, barotropic cyclones typically show an increasing steepness in their flow profiles. This implies that such vortices become barotropically more unstable. This has been confirmed by observations which are further discussed in a companion paper (Kloosterziel & van Heijst 1991). In the present paper the evolutionary process is discussed. It is shown that the observed steepening of the flow profiles is mainly a nonlinear effect due to the advection of relative vorticity by the interior Ekman circulation. The linear Ekman pumping law is found to be a good approximation in the core of the vortices for O(1) Rossby numbers but at larger radii the Ekman pumping is stronger in reality. Free-surface effects are shown to have a broadening effect, which can balance the nonlinear steepening if the Rossby number becomes sufficiently small or the Froude number sufficiently large. In addition it is shown that for the classical spindown problem with a free surface no expansion in the Froude number needs to be introduced.

The disturbance field induced due to a harmonic point source consists of discrete eigenmodes and a continuous spectrum; these are studied by using generalized Fourier transform techniques. For a supersonic boundary layer, there exist seven branches of the continuous spectrum in the complex wavenumber space, four of which (two acoustic waves, one vorticity wave and one entropy wave) contribute to the flow field downstream of the source. The discrete eigenmodes spring off from these branches at some critical Reynolds numbers. The results for Mach 2 and 4.5 boundary layers show that the receptivity coefficients for the stable discrete modes are much larger than that for the unstable mode. Therefore, the flow very near the source is dominated by the continuous spectrum and the stable discrete modes. However, the unstable mode takes over sufficiently far away from the source. It is shown that it is only necessary to consider the first few discrete modes to construct the solution. Calculations also show that, in a supersonic boundary layer, upstream influence from a localized disturbance is minimal.

The weakly nonlinear, high-Reynolds-number triple-deck theory of Smith (1979) is applied to Blasius flow over a compliant wall. Attention is concentrated on Tollmien–Schlichting (TS) disturbance waves. We consider wall models of the Carpenter–Garrad type, modified to cater for three-dimensional disturbances, and allowing for the effects of nonlinear wall curvature. Supercritical equilibrium-amplitude states are possible for TS waves in a rigid-wall boundary layer, as is well known (see for example Smith 1979; Hall & Smith 1984). It is found that judicious choice of wall parameters can dramatically alter the nonlinear stability properties of TS waves in the boundary layer over a compliant wall: waves that are linearly damped may become nonlinearly unstable. Excellent agreement is obtained with rigid-wall results of Hall & Smith (1984).

The spreading of a drop of viscous fluid on a horizontal surface by capillarity has been studied by a number of authors. Different hypotheses have been advanced for the crucial questions of the contact angle at the moving rim of the drop. It is argued that there is one model that agrees with experiments and is economical in its hypotheses. On the basis of this model, the spreading rate is calculated for small static contact angles and for complete wetting (zero contact angle). The rates are also found when the spreading depends partially or dominantly on gravity.

A numerical investigation is made of the effects of compressibility on three-dimensional thermal convection in a basally heated, highly viscous fluid spherical shell with an inner to outer radius ratio of approximately 0.55, characteristic of the Earth's whole mantle. Compressibility is implemented with the anelastic approximation and a hydrostatic adiabatic reference state whose bulk modulus is a linear function of pressure. The compressibilities studied range from Boussinesq cases to compressibilities typical of the Earth's whole mantle. Compressibility has little effect on the spatial structure of steady convection when the superadiabatic temperature drop across the shell ΔTsa is comparable to a characteristic adiabatic temperature. When ΔTsa is approximately an order of magnitude smaller than the adiabatic temperature, compressibility is significant. For all the non-Boussinesq cases, the regular polyhedral convective patterns that exist at large ΔTsa break down at small ΔTsa into highly irregular patterns; as ΔTsa decreases convection becomes penetrative in the upper portion of the shell and is strongly time dependent at Rayleigh numbers only ten times the critical Rayleigh number, 〈Ra〉cr. Viscous heating in the compressible solutions is concentrated around the upwelling plumes and is greatest near the top and bottom of the shell. Solutions with regular patterns (and large ΔTsa) remain steady up to fairly high Rayleigh numbers (100〈Ra〉cr), while solutions with irregular convective patterns are time dependent at similar Rayleigh numbers. Compressibility affects the pattern evolution of the irregular solutions, producing fewer upwelling plumes with increasing compressibility.