We examine a three-dimensional rivulet flowing down a vertical plane. There exists a basic state with fully developed, unidirectional flow and straight contact lines. In the absence of contact-angle hysteresis the slope of the contact angle versus contact-line speed relationship measures the mobility of these contact lines. The stability characteristics of flat rivulets subject to long wave disturbances are examined using lubrication theory. We find that kinematic-wave instabilities are predicted for wide rivulets or rivulets with rather immobile contact lines, while capillary break-up is predicted for narrower rivulets with mobile contact lines. We find for all cases that the expression for the growth rate depends weakly on slip between the liquid and solid near the contact line, but strongly on the shape of the rivulet and the mobility of the contact lines. We discuss in detail the mechanisms by which the contact lines affect the instabilities.

The structure of the vorticity fields in homogeneous turbulent shear flow and various irrotational straining flows is examined using results from direct numerical simulations of the unsteady, incompressible Navier-Stokes equations with up to 128 × 128 × 128 grid points. In homogeneous shear flow, the distribution of the inclination angle of the vorticity vectors and contour plots of two-point correlations of both velocity and vorticity are consistent with the existence of persistent vortical structures inclined with respect to the flow direction. Early in the development of these shear flows, the angle of inclination at which most of these structures are found is near 45°; after the flow develops, this angle lies between 35°–40°. Instantaneous vorticity-vector and vortex-line plots confirm the presence of hairpin vortices in this flow at the two Reynolds numbers simulated. These vortices are formed by the roll-up of sheets of mean spanwise vorticity. The average hairpin leg spacing decreases with increasing Reynolds number but increases relative to the Taylor microscale for developed shear flows. Examination of irrotational axisymmetric contraction, axisymmetric expansion, and plane strain flows shows, as expected, that the vorticity tends to be aligned with the direction of positive strain. For example, the axisymmetric contraction flow is dominated by coherent longitudinal vortices. Without the presence of mean shear, however, hairpin structures do not develop. The simulations strongly indicate that the vorticity occurs in coherent filaments that are stretched and strengthened by the mean strain. When compressed, these filaments appear to buckle rather than to decrease in strength.

Within a granular material stress is transmitted by forces exerted at points of mutual contact between particles. When the particles are close together and deformation of the assembly is slow, contacts are sustained for long times, and these forces consist of normal reactions and the associated tangential forces due to friction. When the particles are widely spaced and deformation is rapid, on the other hand, contacts are brief and may be regarded as collisions, during which momentum is transferred. While constitutive relations are available which model both these situations, in many cases the average contact times lie between the two extremes. The purpose of the present work is to propose constitutive relations and boundary conditions for this intermediate case and to solve the corresponding equations of motion for plane shear of a cohesionless granular material between infinite horizontal plates. It is shown that, in general, not all the material between the plates participates in shearing, and the solutions for the shearing material are coupled to a yield condition for the non-shearing material to give a complete solution of the problem.

Several methods based on matched eigenfunction expansions are developed to study the diffraction of a linear, steady train of waves in finite water depth by a combination of straight depth discontinuities and colinear vortex sheets, where the vortex sheets represent a first approximation to a thin shear layer between regions of dissimilar ambient current velocity. Special attention is paid to the case of a current flowing along a submerged trench, and predictions of reflection and transmission are seen to be highly sensitive to the magnitude and direction of the ambient current.

In order to model the evolution of a solitary wave near an obstacle or over an uneven bottom, the long-wave equations including curvature effects are introduced to describe the deformation and fission of a barotropic solitary wave passing over a shelf or an obstacle. The numerical results obtained from these equations are shown to be in good agreement with an analytical model derived by Germain (1984) in the framework of a generalized shallow-water theory, and with experimental results collected in a large channel equipped with a wave generator. Given the initial conditions, i.e. amplitude of the incident solitary wave, water depth in the deep region, and height of the shelf or the barrier, it is possible to predict the amplitude and number of the transmitted solitary waves as well as the amplitude of the reflected wave, and to describe the shape of the free surface at any time.

An inherently unstable jet impinges upon a leading edge oscillating at controlled frequency and displacement amplitude, giving rise to two coexisting instability waves in the jet: one at the self-excited frequency of the jet; and the other at the controlled frequency of the edge displacement. Correspondingly there is an unsteady loading of the edge at these two frequencies. Simultaneous edge pressure and jet velocity measurements allow insight into the upstream influence, arising from the edge loading, on the jet oscillations. This upstream influence initially distorts the jet at the nozzle exit and causes non-homogeneous phase variations along the streamwise extent of the jet. A simple superposition model which includes upstream-induced velocities and instability-wave velocities effectively simulates these distortions.

The jet oscillations synchronize with the frequency of the controlled edge oscillations for excitation frequencies close to those of the natural jet oscillations. Measurement of the pressure amplitudes on the edge surface shows resonance of the component at the excitation frequency within the synchronization range, and attenuation of the component at the self-excited frequency close to the synchronization range. Depending on the amplitude of edge displacement, synchronization is achieved either by quenching of the self-excited component or by phaselocking of the self-excited component to the excitation frequency. Phase measurements between edge displacement and surface pressure fluctuation allow determination of the direction of energy transfer between the flow and the edge.

Flow-visualization performed simultaneously with pressure measurements gives insight into the relation between impinging vortical structures and pressure fluctuations. Time-sequence photographs allow analysis of the modulation of the flow structure due to coexistence of the self-excited and the externally excited jet instabilities. Vortex coalescence involving vortices of like sense, as well as typical formations of pairs of counter-rotating vortices, are observed. Retardation of the development of the jet vortex pattern occurs when the energy transfer from the flow to the edge is a maximum. At high excitation frequency, the large-scale jet structure recovers to that occurring in absence of edge oscillations.

An experimental, two-layer, stratified model of coastal upwelling was produced at the outer edge of a cylindrical tank having a conical bottom. As in Narimousa & Maxworthy (1985), we find that most of the characteristics of the flow field can be described by the single non-dimensional parameter θ* = g′h0/μ*fλs (where g′ = gδP is the reduced gravity, h0 is the initial depth of the top layer, μ* is the friction velocity applied to the top surface of the fluid, f is the Coriolis parameter and λs is the distance of the stationary position of the upwelling front at the surface from the wall of the tank). For example, we find that the migration rate of the front at the surface, μf is given by μf/μ* ∼ θ−½*; the size λW of the frontal, circular waves is λW/λs ∼ θ*; that these waves first appear at the front when it has moved a distance λi < λs given by λi/λs ∼ θ−½*; and have a final drift velocity μw ≈ 0.5 U in the direction of the applied stress (where U is the mean velocity of the top layer). At low values of θ* complex interactions between the growing waves created intense meandering jets which transported upwelled fluid far from the mean position of the front.

The modal distributions of coherent structures evolving near the nozzle of a circular jet are considered. The effects produced on the instability modes by transverse curvature, flow divergence, inhomogeneous inflow conditions, and the detailed shape of the mean velocity profile, are investigated both theoretically and experimentally. Linear stability analysis applied to a thin shear layer surrounding a large-diameter jet (i.e. a jet whose diameter is large in comparison with a typical width of the shear layer) indicates that many azimuthal modes are equally unstable. An increase in the relative thickness of the shear layer limits the number of unstable modes, and only one helical mode remains unstable at the end of the potential core. The linear model used as a transfer function is capable of predicting the spectral distribution of the velocity perturbations in a jet. This provides a rational explanation for the stepwise behaviour of the predominant frequency resulting from a continuous increase in the jet velocity.

Leading nonlinear interactions generated by waves externally superimposed on an axisymmetric jet are considered theoretically, and the results verified experimentally. The mean flow in the jet loses its axial symmetry whenever the jet is excited simultaneously by two different azimuthal modes of the same frequency. Subharmonic resonance occurs in this flow whenever the conditions warrant such an occurrence, generating azimuthal modes which may not have been present otherwise in this flow. Some of these resonance conditions are explored.

The collapse of cavitation bubbles generates microjets which can cause local plastic deformation on neighbouring solid materials. This deformation takes the form of pitting which will eventually lead to large-scale material erosion. A model is presented which predicts the relative dimensions of the pits as a function of bubble collapse pressure, the shape of the microjet and the mechanical properties of the solid. The high pressures required to cause material deformation are generated by a water hammer mechanism and the solid is taken to have a simple bilinear elastic-plastic response. Measurements on pits produced by both flow and spark-generated cavitation are found to lie within the bounds predicted by the model. Both the measurements and the model suggest that there is a threshold microjet velocity below which no damage occurs, although such behaviour is masked, in practice, by statistical variations.

A numerical solution is provided for flow of a stream of water falling under gravity after emerging from a simple slit orifice in a vertical wall. The classical free-streamline flow with contraction coefficient 0.611 applies in the absence of gravity, or as F → ∞, where F is the Froude number based on the net volume flux and the slit width. We assume here that F is finite, but find that the flow exists only for F [ges ] 0.496, with initial backflow at the upper stream surface for 0.496 [les ] F [les ] 0.578. The limiting flow at F = 0.496 has a stagnation point at the upper edge.

The three components of the average temperature dissipation have been measured using a pair of parallel cold wires in an approximately self-preserving turbulent boundary layer. The mean square value of θ,x the temperature derivative in the longitudinal direction, is determined mainly by the use of Taylor's hypothesis, following direct verification of this hypothesis at a few locations in the flow. Mean square values of θ,y and θ,z, the temperature derivatives in directions normal to the flow, were estimated mainly from the curvature of spatial temperature autocorrelations. In the outer layer, the measurements indicate that $\overline{\theta^2}_{,z} > \overline{\theta^2}_{,y} > \overline{\theta^2}_{,x}$, and the resulting distribution for dissipation leads to a good closure of the $\frac{1}{2}\overline{\theta^2}$ budget. In the near-wall region the measurements indicate that $\overline{\theta^2}_{,y} > \overline{\theta^2}_{,z} > \overline{\theta^2}_{,x}$. The ratios $\overline{\theta^2}_{,y}/\overline{\theta^2}_{,x} $ and $\overline{\theta^2}_{,z}/\overline{\theta^2}_{,x}$ are as large as 13 and 7 respectively at y+ = 12, underlining the strong anisotropy in this region. The behaviour of the turbulent diffusion, estimated by difference, provides reasonable support for the accuracy of the near-wall temperature-dissipation measurements. Using existing data of near-wall distributions of the turbulent energy and of its dissipation rate, the timescale for the turbulent-energy dissipation is found to be approximately equal to that for the temperature dissipation.

The flow of a liquid with a free surface over a weir in a channel is calculated numerically for thin weirs in channels of various depths, and for broad-crested weirs in channels of infinite depth. The results show that the upstream velocity, as well as the entire flow, are determined by the height of the free surface far upstream and by the geometry of the weir and channel, in agreement with observation. The discharge coefficient is computed for a thin weir, and a formula for it is given that applies when the height of the weir is large compared to the height of the upstream free surface above the top of the weir. The coefficients in this formula are close to those found empirically.

The results of a theoretical investigation of the low-speed motion of a volatile spherical particle in its saturated vapour at arbitrary values of the Knudsen number, evaporation coefficient and tangential momentum accommodation coefficient are presented. The problem is solved on the basis of the Bhatnagar, Gross & Krook (1954) linearized kinetic equation with the model collision integral. The variational method developed for the solution of the axisymmetric problems (Cercignani & Pagani 1968) is employed to calculate the drag. Phase changes on the particle surface have been shown to reduce the drag by an order depending on the Knudsen number. Incomplete accommodation of molecules colliding with the particle also results in the reduction of the drag over the whole range of Knudsen numbers.

The influence of a non-uniform medium on the Benjamin-Feir instability of weakly nonlinear deep-water waves has been investigated, and an approach via a suitable nonlinear Schrödinger equation was adopted. For the derivation of the relevant cubic Schrödinger equation. the approach of Yuen & Lake (1975) was followed and an applicable dispersion relation and energy equation was derived by the averaged Lagrangian technique. With the assumption that the lengthscale of current variation is much greater than the lengthscale of the wavepacket, a cubic Schrödinger equation with slowly varying coefficients is obtained. Three different examples of non-uniform media are treated: (i) waves on a current with variation along the stream; (ii) waves on a shear current; and (iii) long deep-water gravity waves interacting with shorter waves.

The effect of surface roughness on the transition form regualr (RR) to Mach reflection (MR) over straight wedges in pseudo-steady flows was investigated both experimentally and analytically. A model for predicting the RR \rightleftarrows$ MR transition in the (Mi, θw)-plane was developed (Mi is the incident shock wave Mach number and θw is the reflecting wedge angle). Its valdity was checked agnainst experimental results. Since the experimental results are limited to the ranges 1 < Mi < 2 and surface roughness heights of 0 < ε [les ] 0.2 cm, the proposed model is applicalbe to these ranges only. In the proposed model (the boundary-layer displacement-thickness model), the RR \rightleftarrows$ MR transition is related to the boundary - layer thickness which in turn depneds on the surface roughness.

The equilibrium shape and stability of menisci formed at the contact line between two vertically aligned cylinders were investigated by developing a general bifurcation analysis from the classic equation of Young-Laplace. It was found that the maximum amount of liquid that can be held at the contact line is determined by the existence of a bifurcation of the equilibrium solutions. The onset of instability is characterized by a translationally symmetric bifurcation that always precedes the instability to asymmetric perturbations. The maximum stable liquid retention is a strong function of the ratio of gravitational to surface-tension forces, indicating that gravity acts as a destabilizing force. The effect of contact angle on the maximum liquid retention is more complex: when the gravitational effects are small, an increase in contact angle results in a decrease in liquid retention; on the other hand, when the gravitational effects are appreciable, a maximum value of the liquid retention is obtained for intermediate values of the contact angle.

We consider the response of a Newtonian fluid, saturating the pore space of a rigid isotropic porous medium, subjected to an infinitesimal oscillatory pressure gradient across the sample. We derive the analytic properties of the linear response function as well as the high- and low-frequency limits. In so doing we present a new and well-defined parameter Λ, which enters the high-frequency limit, characteristic of dynamically connected pore sizes. Using these results we construct a simple model for the response in terms of the exact high- and low-frequency parameters; the model is very successful when compared with direct numerical simulations on large lattices with randomly varying tube radii. We demonstrate the relevance of these results to the acoustic properties of non-rigid porous media, and we show how the dynamic permeability/tortuosity can be measured using superfluid 4He as the pore fluid. We derive the expected response in the case that the internal walls of the pore space are fractal in character.

A numerical analysis has been performed of three-dimensional time-dependent solutions which bifurcate supercritically from two-dimensional convection-roll solutions at the onset of the oscillatory instability. The bifurcating solutions describe a periodic shifting forward and backward of the convection rolls and lead to a strong deformation of the rolls as the Rayleigh number increases. Since the bifurcating solution is stable in the form of a travelling wave, the computational expense can be reduced by assuming a moving coordinate. Travelling-wave solutions have been computed in the case of rigid boundaries as a function of the Prandtl number and of the two basic wavenumbers αx, αy of the problem. The onset of oscillations reduces the heat transport in comparison with that of two-dimensional rolls because the occupation of a new degree of freedom of motion by the oscillation reduces the energy of the heat-transporting component of convection. A limited stability analysis of finite-amplitude travelling waves has been performed and the onset of an asymmetric mode of oscillations is determined as a function of the parameters of the problem. This mode appears to be identical with a mode that was observed in the numerical simulations of Lipps (1976) and McLaughlin & Orszag (1982).

A boundary-layer approximation is used to describe the convective regime in a laterally heated vertical slot at large Prandtl numbers. The determination of the core flow requires the solution of the vertical boundary-layer equations in a rectangle, subject to appropriate boundary conditions on each of the four walls. Solutions based on a spectral decomposition in the vertical direction allow a comparison with experimental and numerical results, and an appraisal of an approximate solution frequently used as a basis for stability studies. Both the numerical results and an approximate stability argument lead to a simple criterion for the appearance of multiple rolls in the slot which appears to be in good agreement with experiments.