The stability of stratocumulus clouds with strong evaporative cooling effects is explored in laboratory simulations. Two fluids, initially separated by a thin horizontal plate, contain mixtures of water, alcohol and glycol which have a strongly nonlinear density as a function of mixture ratio. Initially, the fluid below the plate is more dense than that above the plate. When the plate is suddenly withdrawn, the turbulence in its wake mixes the two fluids together, producing mixtures with densities greater than that of either initial fluid. It is found that the system is unstable to strong perturbations only in cases of relatively large buoyancy reversal. The system is stable to strong perturbations if the buoyancy reversal is comparable to or less than the initial stratification. A simple model is presented to explain the results.

The instability of a compressible flow past a wedge is investigated in the hypersonic limit. Particular attention is paid to Tollmien-Schlichting waves governed by triple-deck theory though some discussion of inviscid modes is given. It is shown that the attached shock can have a significant effect on the growth rates of Tollmien–Schlichting waves. Moreover, the presence of the shock allows for more than one unstable Tollmien–Schlichting wave. Indeed an infinite discrete spectrum of unstable waves is induced by the shock, but these modes are unstable over relatively small but high frequency ranges. The shock is shown to have little effect on the inviscid modes considered by previous authors and an asymptotic description of inviscid modes in the hypersonic limit is given.

The resonant excitation of weakly nonlinear, standing edge waves of frequency ω and longshore wavenumber k by a normally incident, non-breaking gravity wave of frequency 2ω and shoreline amplitude a to on a bottom that descends smoothly from a shoreline depth of zero and slope σ to an offshore depth h∞ is calculated for ka [Lt ] σ [Lt ] kh∞ [Lt ] 1. The analysis generalizes hose of Guza & Bowen (1976) and Rockliff (1978), which assume uniform slope and perfect reflection, and culminates in a pair of evolution equations for the slowly varying, quadrature amplitudes (or, equivalently, amplitude and phase) of the edge wave. Weak, linear damping (which implies imperfect reflection) is incorporated, and the resulting fixed points and bifurcation points of the evolution equations are determined. It is shown that the solution for prescribed initial conditions must tend to one of the stable fixed points, which correspond to an edge-wave amplitude of either zero or O((σa/k)½), depending on whether the damping exceeds or is inferior to a certain critical value. The restriction σ [Lt ] kh∞ is relaxed for the special depth profile h/h∞ = 1 −exp (−σx/h∞), for which the inviscid, shallow-water equations admit exact solutions. These solutions serve to validate the asymptotic (σ/kh↓O) approximations for arbitrary depth profiles.

The linear reflection of an obliquely incident gravity wave of frequency ω from a gently sloping beach of shoreline slope σ and characteristic length l is determined for σ [Lt ] 1 [Lt ] ω2l/g. An asymptotic (σ↓0), inviscid approximation that is uniformly valid in the shallow-water domain is matched to Keller's (1958) geometrical-optics approximation for non-shallow water. An exact solution is obtained for the profile h = σl[1−exp (−x/l)] in the shallow-water domain and used to test the asymptotic approximation. The absence of viscosity implies perfect reflection. A model that incorporates both small viscosity and small capillarity predicts a fixed contact line and the reflection coefficient |R| = exp [−πσ−2g−1(2νω3)½], where ν is the kinematic viscosity. These predictions are in qualitative agreement with the experimental results of Mahony & Pritchard (1980).

It is well known that when a high subsonic (Mach number > 0.7) high Reynolds number (Re > 2 × 105) jet is directed normal to a wall intense discrete frequency sound waves called impingement tones are emitted. This phenomenon has been studied by a number of investigators in the past. It is generally accepted that the tones are generated by a feedback loop. Despite this general agreement critical difference in opinion as to how the feedback is achieved remains unresolved. Early investigators (e.g. Wagner 1971; Neuwerth 1973, 1974) proposed that the feedback loop is closed by acoustic disturbances which propagate from the wall to the nozzle exit inside the jet. Recent investigators (e.g. Ho & Nosseir 1981; Umeda et al. 1987), However, believed that the feedback is achieved by sound waves propagating outside the jet. In this paper a new feedback mechanism is proposed. It is suggested that the feedback is achieved by upstream-propagating waves associated with the lowest-order intrinsic neutral wave modes of the jet flow. These wave modes have well-defined radial and azimuthal pressure and velocity distributions. These distributions are dictated by the mean flow of the jet exactly as in the case of the well-known Kelvin-Helmholtz instability waves. The characteristics of these waves are calculated and studied. These characteristics provide a natural explanation of why the unsteady flow fields of subsonic impinging jets must be axisymmetric, whereas those for supersonic jets may be either axisymmetric or helical (flapping). In addition they also offer, for the first time, an explanation as to why no stable impingement tones have been observed for (cold) subsonic jets with Mach number less than 0.6. Furthermore, the new model allows the prediction of the average Strouhal number of impingement tones as a function of jet Mach number. The predicted results compare very favourably with measurements. For subsonic jets the pressure and velocity field of these upstream-propagating neutral waves are found to be confined primarily inside the jet. This is in agreement with the observations of Wagner (1971) and Neuwerth (1973, 1974) and their contention that the feedback disturbances actually propagate upstream inside the jet.

The higher-order boundary-layer solution for an impulsively started circular cylinder with uniform velocity and for an exponentially accelerating cylinder in incompressible, relatively high-Reynolds-number flow of short duration are considered. A perturbation method is employed to linearize the two-dimensional vorticity transport equation by a double series expansion with respect to the Reynolds number and the time. A matched asymptotic expansion is carried out to define the proper boundary conditions between the viscous and inviscid layers for the linearized first-, second-, and third-order boundary-layer equations. Singularities appear in the higher-order approximate solutions to the viscous displacement velocities and skin frictions, which coincide with the singularity of the first-order approximate solution. These singularities have alternating signs and increasing magnitudes, thus attempting to remove the effects of the singularity of the lower-order solution. However, this futile attempt at removing a singularity by superposing even stronger singularities makes the solution worse around the singularity, which shows that the singularity is an artifact of the thin-boundary-layer approximation.

A number of experiments have indicated that the behaviour of longitudinal vortices in the sublayer of a turbulent boundary layer has a significant effect on the equilibrium of the whole flow field, but the formation of such longitudinal vortices remains unclarified. In the present paper, paying attention to the random turbulent motions normal to the wall which induce a dynamic instability causing the generation of the regular longitudinal vortices, a turbulence model is introduced to analyse the generation of the longitudinal vortices in the sublayer of a turbulent boundary layer.

By integrating the results of the analysis with previous experimental results for the behaviour of developing longitudinal vortices, a feedback loop linking regular and irregular motions in the turbulent boundary layer is presented.

It has been experimentally confirmed that even at low Reynolds number, regular longitudinal vortices are formed when a disturbance is applied to a boundary layer. As for the stability region described in terms of the intensity of the disturbance and the wavenumber of the longitudinal vortices, no conflict between analysis and experiment is observed. Using the thermodynamic argument of irreversible processes the formation of longitudinal vortices close to the wall by the turbulence in the boundary layer is shown to be physically reasonable.

The coexistence of migrating alternate (free) bars, spontaneously developing in erodible channels as a result of an instability process, with steady point bars, forced by curvature in meandering reaches of rivers, is investigated theoretically.

A perturbation expansion is set up in terms of two dimensionless small parameters, ε and ν respectively, describing free and forced perturbations. The effect of mixed interactions at O(ν2ε½) is found to be responsible for the damping and slowing down of free bars as channel curvature increases. The theory allows us to determine the threshold value of channel curvature above which free bars are suppressed as a function of meander wavenumber for given flow and sediment characteristics. The minimum channel sinuosity for free bar suppression is found to be associated with the resonant wavenumber range of Blondeaux & Seminara (1985). Theoretical predictions compare satisfactorily with experimental observations by Kinoshita & Miwa (1974). The theory also suggests that free bars may appear again in a more advanced stage of meander development in accordance with field observations by Kinoshita (1961).

A Fourier-integral method is developed to obtain transient solutions to potential wavemaker problems. This method yields solutions for wavemaker velocities which need not be given as powers of time. The results are compared with known small-time and local solutions. Examples considered include ramp, step, and harmonic wavemaker velocities. As time becomes large, the behaviour near the wave front is derived for the impulsive wavemaker, and for the harmonic wavemaker it is shown that the steady-state solution is recovered. The solution for a wavemaker velocity given as a Fourier cosine series compares favourably with available experimental results. Capillary effects are included and nonlinear effects are discussed.

An amplitude equation for weak interactions of waves is derived to describe the time evolution of disturbances on an interface between fluids of differing densities, with rigid upper and lower boundaries. This equation is analogous to the Zakharov equation for water waves and is used to investigate the stability of a periodic wave. It is found that, for small wave steepnesses, the instabilities are due to resonant quartets with perturbation wavenumbers of the order of, or less than, that of the main wave. A second instability is found for large perturbation wavenumbers and moderately high wave steepnesses. This is restricted to the case when the Boussinesq parameter is small. It is shown that this is a Kelvin–Helmholtz instability caused by a wave-induced jump in the fluid velocity across the interface.

Flows transporting material between nearby moving surfaces are ubiquitous in machinery of all scales and with a variety of geometries. Here a general derivation is given of the effective two-dimensional mixing process in a narrow gap for a solute or miscible fluids. Explicit formulae are given for the shear dispersion tensor in laminar and turbulent (logarithmic velocity profile) flows. It is shown that if improved mixing is required, then the optimum direction for additional boundary motion or stress is at right angles to the primary flow direction.

High-Reynolds-number steady flow in an annular pipe which encounters a shallow axisymmetric expansion or indentation in the walls is studied using interactive boundary-layer theory. The flow upstream of the indentation (x < 0) is fully developed; the ratio of the shear rate on the outer wall to that on the inner wall is denoted by ρ (0 < ρ < 1): similarity solutions are found for the case where the wall perturbations are proportional to $x^{\frac{1}{3}}$. The solution is unique in a constriction, when the pressure gradient (represented by a parameter b) is favourable (b < 0). In an expansion, however, with an adverse pressure gradient, three different solutions are found if b exceeds a critical value bc. When ρ ≠ 1, one of these solutions, representing a flow that is attached on the inner wall and separated (i.e. has negative wall shear) on the outer, is a continuation of the unique doubly attached flow at small b. The other two, one separated on the inner and not the outer wall and the other separated on both walls, arise from a saddle-node bifurcation at b = bc. The doubly separated flow is never stable, as observed in diffusers. In the case of a planar channel (ρ = 1) symmetry is restored, and the non-uniqueness arises through a supercritical pitchfork bifurcation. This agrees with previous computations on channel flow, but not with Jeffery-Hamel flow, for which the bifurcation is subcritical.

The stability of core-annular flow in vertical pipes is analysed using the linearized theory of stability. In previous studies instabilities due to interfacial friction, interfacial tension and Reynolds stresses in the bulk fluid were identified and associated with observed instabilities. In this study we include and analyse the effects of gravity. In one case gravity opposes and in the other aids the applied pressure gradient. Some preliminary results from our experiments are also presented. The prediction of stability for perfect core-annular flow in a carefully selected window of parameters is verified for the case of free fall in which the applied pressure gradient vanishes.

Convection experiments were conducted in a porous slab 3 m in diameter and 30 cm thick, using two quite different media: a filling of rubberized curled coconut fibre and of clear polymethylmethacrylate beads. The second experiment involved the successful use of a visualization scheme for the flows at the upper boundary. Convection began in a hexagonal pattern with a slight tendency to form into rolls, but became very complex, irregular and three-dimensional at higher Rayleigh numbers, without developing any obvious temporal instabilities. Above a Rayleigh number of 1000 a significant number of dendritic downwellings appeared, where smaller downwellings seemed to feed into larger areas such that the whole complex may have converged into a single downwelling plume. The visualization provides direct confirmation that the lateral scale of the convection decreases with increasing Rayleigh number, approximately as ([Ascr ] + C)−0.5.

Nusselt number versus Rayleigh number curves were obtained for both experiments. The only feature they have in common is a central section where the slope on a log/log graph is slightly over 0.5. On the graph from the first experiment, this section is preceded by a slope close to 1 and followed by a slope close to 0.33. The temperature measured at a point in the fill 25 mm below the top boundary was unsteady at conditions representative of the upper two segments of the graph; sensitivity was insufficient to measure fluctuations at lower temperature differences. The Nusselt number for the bead fill jumps upward just above onset (where [Ascr ] = 4π2), rapidly settles to a slope of 0.52, and then gradually breaks upward again to a slope of greater than 1. Increases in conductivity and permeability close to the boundary are not a large enough fraction of the boundary-layer thickness to cause this. A new phenomenon, lateral thermal dispersion, appears to be responsible. It occurs because there is no constant separation distance between adjacent channels in a bead fill. Thermal exchange in the junction pores exceeds the average if the flow is fast enough, especially when the fluid is more conductive than the beads.

A simple boundary-conduction theory can be matched to the uncontaminated results. It is based on relative scaling of the residence time of fluid in the boundary layer, and predicts Nusselt number growth as the 0.55 power of Rayleigh number toward the high values typical of major geothermal areas.

A nonlinear Schrödinger equation, describing the evolution of a weakly nonlinear short gravity wavetrain riding on a longer finite-amplitude gravity wavetrain, is derived. This equation is then used to predict the steady envelope of the short wavetrain relative to the long wavetrain. It is found that approximate analytical solutions agree very well with numerical solutions over a realistic range of wave steepness. The solutions are compared with corresponding studies of the modulation of linear short waves by Longuet-Higgins & Stewart (1960) and Longuet-Higgins (1987). We find that the effect of the nonlinearity of the short waves is to increase the modulation of their wavenumber, significantly reduce the modulation of their amplitude, and reduce the modulation of their slope when compared with the predictions of Longuet-Higgins (1987) for linear short waves on finite-amplitude long waves. The question of the stability of these steady solutions remains open but may be addressed by solutions of this nonlinear Schrödinger equation.

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

Observations of the cavity produced by a droplet of rain falling on a plane water surface show that the cavity often takes the form of an inverted cone. In the present paper we discuss a simple analytic expression for the corresponding fluid flow.

The main source of rain noise has been found to occur when the flow entrains an air bubble at the vertex of the cone. In this paper we show that the magnitude of the sound produced by the fluid flow in the present model is consistent with observation.

A completely non-intrusive technique, utilizing synchronized flow visualization and one-point, two-component laser-Doppler velocimetry, has been developed to examine the vortical structure of a homogeneous plane shear layer at a moderate Reynolds number. Results include ensemble-averaged velocity and vorticity maps and, derived from these data, zone-averaged statistics of the basic structure, at 6.2 initial wavelengths downstream of the splitter-plate trailing edge. The velocity field data paint a clear picture of the processes involved in the engulfment or entrainment of free-stream irrotational fluid by the vortices, and the evolution of the material interface separating fluid from the two free streams. The ensemble-averaged vorticity distribution is seen to be of an elliptical cross-section, closely resembling that of the Stuart vortex for a vorticity distribution parameter of 0.4. The effect of jitter which, in the case of zone averages is effectively removed, is assessed by comparison to conventional mean flow properties. It is found that vortex jitter greatly influences u′ and Ruu and, in confirmation of previous results, that a passive vortex does not contribute to the production of large-scale Reynolds stress.

Quasi-geostrophic β-plane motion of a homogeneous liquid over topography is considered for situations in which there is a time-periodic forcing of a zonal current. Such an oscillatory current generates a topographic Rossby wave response that has a complicated, but periodic, temporal structure. The linear solution shows resonances at all integer values of the β-parameter. The nonlinear analysis demonstrates that for weak friction and forcing, the resonances are bent and multiple equilibria of the time-dependent Rossby wave states are possible in certain parameter ranges. While the basic forced flow in the absence of topography has no time-mean, the nonlinear amplitude equations show that a mean retrograde (westward) Eulerian zonal flow is generated in the interactions of the forced flow with the topography. This result is in agreement with a previous theory of Samelson & Allen, constructed for strongly nonlinear flow over a series of asymptotically long ridges. However, in contrast to the behaviour of their amplitude equations for certain parameter settings, the nearresonant weakly nonlinear model for more or less isotropic bottom topography appears non-chaotic for all accessible parameter values.

The unsteady expansion of an ideal gas into a vacuum is studied in one-dimensional planar and spherical geometries. The free-molecular expansion of a Maxwell-distributed gas is compared to the continuum expansion of a perfect gas with $\gamma = \frac{5}{3}$. Time histories of density, temperature, and wall pressure (i.e. pressure on a wall surface oriented normal to the flow) are given at four near-field locations, and the approach to far-field behaviour is illustrated. In the free-molecular limit, closed-form expressions for the wall pressure, translational temperature, and fluxes of momentum, kinetic energy, and thermal energy have been obtained in addition to previously published results for density and velocity. The density and dynamic fluxes are observed to decay more rapidly in the tails of continuum pulses than in free-molecular pulses. The reverse is true for wall pressure, which decays less rapidly in continuum flow. Translational temperature, in the free-molecular case, rises discontinuously upon pulse arrival, and, at long times approaches $\frac{2}{3}$ for planar flow and tends to zero for spherical flow. Continuum thermodynamic temperature pulses, on the other hand, rise and fall in simple relation to continuum density. The far-field peak wall pressure in both Knudsen-number limits is found to decrease in inverse (or inverse cubic) proportion to the distance from the initial planar (or spherical) region. This result for the spherical case is at odds with the experiments of Ahrens, Allen & Kovach (1971) which indicate a more rapid (ξ−3.5) fall-off of peak overpressure with distance from a point source in a vacuum.