Numerical solutions for fully nonlinear two-dimensional irrotational free-surface flows form the basis of this study. They are complemented and supported by a limited number of experimental measurements. A solitary wave propagates along a channel which has a bed containing a cylindrical bump of semicircular cross-section, placed parallel to the incident wave crest. The interaction between wave and cylinder takes a variety of forms, depending on the wave height and cylinder radius, measured relative to the depth. Almost all the resulting wave motions differ from the behaviour which was anticipated when the study began. In particular, in those cases where the wave breaks, the breaking occurs beyond the top of the cylinder. The same wave may break in two different directions: forwards as usual, and backwards towards the back of the cylinder. In addition small reflected waves come from the region of uniform depth beyond the cylinder. Experimental results are reported which confirm some of the predictions made. The results found for solitary waves are contrasted with the behaviour of a group of periodic waves.

A straightforward method that yields explicit transmission amplitudes is presented for Kelvin wave scattering by topography whose isobaths are parallel sufficiently far from the vertical, but not necessarily planar, wall supporting the incident wave. These results are obtained by first restricting attention to the low-frequency limit in which the flow splits naturally into three regions: an outer-x region containing the incident and transmitted Kelvin waves, an outer-y region containing outwardly propagating long topographic waves and an inner quasi-steady geostrophic region whose structure follows from earlier time-dependent analyses. The present analysis is further simplified by approximating general smooth features by stepped profiles with no restriction on the size, number or order of steps. Various qualitative results on the transmission amplitudes and flow fields are deduced from the explicit solutions and results are given on orthogonality, completeness and direction of propagation of the scattered long waves.

Material line and surface elements transported in a turbulent velocity field increase in length or area at an exponential rate. In this paper we investigate how the stretching rates are related to the statistical properties of the velocity field both analytically and numerically in simple models of turbulence. In a Gaussian model the statistics exhibit time-reversal invariance. We demonstrate that, as pointed out by Kraichnan (1974), this leads to equality of line and area stretching rates. We also construct a model which violates the time-reversal property and splits the values of the rates for lines and surfaces. The sign of the splitting depends on the sign of the time-reversal breakdown.

Experiments have been performed on the stability of buoyancy-driven flows of a high-Prandtl-number fluid in an inclined rectangular enclosure. Visualization of the stable planform of convection for various Rayleigh numbers and inclination angles is provided by a temperature-sensitive liquid crystal and gold-coated film heater assembly which serves as the lower surface of the enclosure. This assembly produces a nearly constant heat flux surface with a thermal conductivity of the same order as that of the test fluid. The results indicate that for large angles of inclination from the horizontal a steady transverse roll(s) structure is stable. As the angle of inclination is decreased steady longitudinal rolls replace the transverse roll(s) and for low angles a steady square-cell convection planform is observed. A region of unsteady wavy longitudinal rolls is also observed at sufficiently high Rayleigh numbers for low to moderate angles of inclination. In general the wavenumber of the longitudinal rolls increases with angle of inclination from the horizontal. Two distinct types of instability mechanisms are observed which modify the wavenumber of the longitudinal rolls: a cross-roll instability, which is a disturbance perpendicular to the original roll axis; and a pinching mechanism which combines two neighbouring longitudinal roll pairs into a longer wavelength roll pair.

The response of mass transfer to a small mass sink to hydrodynamic fluctuations in the concentration boundary layer has been calculated as a function of frequency. The dimensionless local flux was expressed as a series expansion of the dimensionless local diffusion layer thickness η and the dimensionless local characteristic frequency ξ in the low frequency range, and as the asymptotic power law $\xi^{-\frac{1}{3}}$, in the high frequency range. The two solutions were shown to overlap fairly well for 6 [les ] ξ [les ] 13. The overall transfer function over the whole mass sink area involves a spatial distribution for which the low-frequency approximation applies at the upstream end and the high-frequency approximation applies downstream. The average response at frequency f varies as f−1.

These theoretical predictions were tested electrochemically by using a rotating disk. The modulated limiting diffusion current due to a fast redox reaction at small circular microelectrodes embedded in the disk was measured as a function of the frequency of the modulation of the disk angular velocity.

Experimental investigations were carried out on the coherent structures of turbulent boundary layers in flow with adverse pressure gradient and, in the vicinity of separation, extensive visual observations using hydrogen bubble technique have been performed. In a flow with adverse pressure gradient the size of the structures are larger, therefore more details were observed. By a suitable manipulation of the generation of hydrogen bubble time-lines some new results were obtained in observing plan views near the wall. (1) The long streaks downstream along the interface regions between low-speed and high-speed streaks are continually stretching and their velocity may be greater than that of high-speed streaks, and the hydrogen bubbles in the long streaks generally have a longer life. (2) The x, y-vortices (streamwise) were also observed along the interface regions between high-speed and low-speed streaks. (3) The z-vortices (transverse) were observed at the front of the high-speed regions.

The aim of the present work is to investigate the existence regions of the different flow patterns exhibited by a liquid flowing down an inclined plane for a wide range of physical properties of the fluid (particularly surface tension and viscosity which were found to have the greatest influence). A model that predicts the decay frequency of oscillating or pendulum rivulets is presented. From this model, a stability criterion for the onset of oscillating rivulet flow is derived. Although the model does not contain any freely adjustable parameters, it shows good agreement with experimental measurements of rivulet decay frequency and of the transition point to pendulum rivulet. The transitions between different flow regimes are expected to cause drastic changes in heat and mass transfer rates between the liquid and the solid surface or between the liquid and the surrounding gaseous phase.

A theory is presented which describes the propagation of a ring wave on the surface of a flow which moves with some prescribed velocity profile. The problem is formulated in suitable far-field variables (which give the concentric KdV equation for a stationary flow), but allowance is made for the fact that the wavefront is no longer circular. The leading order of this small-amplitude long-wave theory reduces to a generalized Burns condition which is used to determine the shape of the wavefront. This condition is written as \[ (h^2+h^{\prime 2}\int^1_2dz/[F(z, \theta)]^2=1, \] where F(z, θ) = -1 + {U(z) − c} (h cos θ − h′ sin θ), U(z) is the velocity profile, c is a parameter and the local characteristic coordinate for the wave is ξ = rh(θ) − t. (The Burns condition is interpreted in terms of the finite part of the integral in order to allow the possibility of a critical layer where F(zc, θ) = 0, 0 < zc < 1.) The wavefront is represented by r = constant /h(θ). A model boundary-layer profile, which gives rise to a critical-layer solution, is chosen for U(z). The role of this critical-layer solution, and the general question of upstream propagation, is then examined by constructing a wavefront which is continuous from the downstream to the upstream side. Solutions are presented which demonstrate that a critical layer never appears and so upstream propagation is necessary. These solutions (for various values of surface speed and boundary-layer thickness) are one branch of what we might term the singular solution of the differential equation for h(θ). The other branch corresponds to a solution which has a critical layer for all θ, which would seem to be unphysical since this solution is not an outward propagating ring wave.

At the next order we obtain the equation which describes the dominant contribution to the surface wave, in this approximation. The equation is a new form of Korteweg–de Vries equation; the novel feature is the dependence on the polar angle, θ. This equation is not analysed in any detail here, but the connection with plane waves over a shear flow, and with concentric waves in the absence of shear, is made.

The equations for the two-dimensional motion of a layer of uniform vorticity in an incompressible, inviscid fluid are examined in the limit of small thickness. Under the right circumstances, the limit is a vortex sheet whose strength is the vorticity multiplied by the local thickness of the layer. However, vortex sheets can develop singularities in finite time, and their subsequent nature is an open question. Vortex layers, on the other hand, have motions for all time, though they may develop singularities on their boundaries. Fortunately, a material curve within the layer does exist for all time. Under certain assumptions, its limiting motion is again the vortex sheet, and thus its behaviour may indicate the nature and possible existence of the vortex sheet after the singularity time. Similar asymptotic results are obtained also for the limiting behaviour of the centre curve as defined by Moore (1978). By examining the behaviour of a sequence of layers, some physical understanding of the formation of the curvature singularity for a vortex sheet is gained. A strain flow, induced partly by the periodic extension of the sheet, causes vorticity to be advected to a certain point on the sheet rapidly enough to form the singularity. A vortex layer, however, simply bulges outwards as a consequence of incompressibility and subsequently forms a core with trailing arms that wrap around it. The evidence indicates that no singularities form on the boundary curves of the layer. Beyond the singularity time of the vortex sheet, the limiting behaviour of the vortex layers is non-uniform. Away from the vortex core, the layers converge to a smooth curve which has the appearance of a doubly branched spiral. While the circulation around the core vanishes, approximations to the vortex sheet strength become unbounded, indicating a complex, local structure whose precise nature remains undetermined.

A tilted Gaussian formula is given which approximately models the concentration distribution at moderate-to-large times after discharge in steady plane parallel flows.

A numerical study of the transition from steady to oscillatory streamwise-oriented vortices in fully developed rotating channel flow is presented. The principal results are obtained from three-dimensional, spectral simulations of the incompressible time-dependent Navier–Stokes equations. With increasing Reynolds number, two transitions that cause the steady, periodic array of two-dimensional vortices (roll cells) to develop waves travelling in the streamwise direction are discovered. The linear stability of two-dimensional vortices to wavy perturbations is examined. Associated with the two transitions are two different wavy vortex flows: WVF1 and WVF2. WVF2 is very similar to undulating vortex flow found in curved channel flow simulations (Finlay, Keller & Ferziger 1988) and to wavy Taylor vortex flow. WVF2 is only possible at low rotation rates. In contrast, the dissimilar WVF1 occurs for all rotation rates examined, has shorter streamwise wavelength and, for sufficiently high Reynolds number, has much higher linear growth rate than WVF2. For low rotation rates, WVF1 is similar to curved channel flow twisting vortices, but at higher rotation rates appears dissimilar. Several key qualitative features are discussed that suffice in describing all these wavy vortex flows.

A two-dimensional direct numerical simulation of the natural convection flow of air in a differentially heated square cavity was performed for a Rayleigh number of 1010. The simulation was commenced from isothermal and quiescent conditions and was allowed to proceed to a statistical steady state. Two-dimensional turbulence resulted without the introduction of random forcing. Good agreement of mean quantities of the statistically steady flow is obtained with available experimental results. In addition, the previously proposed (George & Capp 1979) −$\frac{1}{3}$ and $+\frac{1}{3}$ temperature and velocity variations in the buoyant sublayer are confirmed. Other statistics of the flow are consistent with available experimental data. Selected frames from a movie generated from the computational results show very clearly turbulence production via the sequence from initial instability, proceeding through transition, and eventually reaching statistical steady state. Prominent large-scale structures are seen to persist at steady state.

The properties of steady and unsteady ship waves propagating on a free surface discretized with panels are studied. The wave propagation is characterized by an explicit discrete dispersion relation which allows the systematic analysis of the distortion of the wave pattern due to discretization and the derivation of a stability criterion to be met by the numerical algorithm. The conclusions of the study are applied to a panel method used for the computation of steady and time-harmonic free-surface flows past elementary singularities and a ship hull.

A theoretical model is presented for the propagation of long, weakly nonlinear water waves along a channel bounded by sloping sidewalls, on the assumption that h0/w [Lt ] 1, where 2w is the channel width and h0 is the uniform water depth away from the sidewalls. Owing to the non-rectangular channel cross-section, waves are three-dimensional in general, and the Kadomtsev–Petviashvili (KP) equation applies. When the sidewall slope is O(1), an asymptotic wall boundary condition is derived, which involves a single parameter, [Ascr ] = A/h02, where A is the area under the depth profile. This model is used to discuss the development of an undular bore in a channel with trapezoidal cross-section. The theoretical predictions are in quantitative agreement with experiments and confirm the presence of significant three-dimensional effects, not accounted for by previous theories. Furthermore, the response due to transcritical forcing is investigated for 0 < [Ascr ] [les ] 1; the nature of the generated three-dimensional upstream disturbance depends on [Ascr ] crucially, and is related to the three-dimensional structure of periodic nonlinear waves of permanent form. Finally, in an Appendix, the appropriate asymptotic wall boundary condition is derived for the case when the sidewall slope is O(h0/w)½.

Techniques derived from bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh–Bénard problem, Lapwood convection in a two-dimensional saturated porous cavity heated from below. The objective of the study is the explanation of how the multiplicity of solutions observed for lower boundary heating evolves to an apparently unique solution for sidewall heating. The change in boundary conditions from floor to sidewall heating can be effected by smoothly tilting the cavity through 90°. The present study aims to demonstrate the mechanisms that reduce the multiplicity for increasing tilt angle.

The many solutions in the untilted cavity arise from a complex bifurcation structure. The effect of tilting the cavity is to unfold all bifurcations, except those that break the centro-symmetry, and so to create branches disconnected from the primary flow. As the angle of tilt, ϕ, increases most of the limit points at which these branches arise move to higher Rayleigh number Ra. Unexpectedly, for a square cavity, the critical Rayleigh number of the most important limit point (that gives rise to an anomalous stable unicellular flow) is found to be almost independent of the angle of tilt. Moreover the two branches arising at this limit point merge again at higher Rayleigh number to form a continuous closed loop, or isola. As the tilt increases, the upper limit point approaches the lower one until they coalesce at an isola formation point at a critical angle ϕc of 10.72°, the maximum angle at which this anomalous mode can exist. Symmetry-breaking bifurcations destabilize part of the branch and determine a smaller critical angle of 10.23°, the maximum angle for which the anomalous mode is stable. At very small angles of tilt, the path of limit points forms the expected cusp catastrophe in the (ϕ, Ra)-plane and at larger angles the path itself turns back at the isola formation point.

The results reveal as too simplistic the conjecture that the reduction of multiplicity for increasing tilt derives from the movement of disconnected branches to increasingly higher Rayleigh number. The predicted collapse and disappearance of branches at an isola formation point is a further novel mechanism which ensures that only the unicellular primary branch remains at a tilt of 90°, in accord with the expected uniqueness of the flow in a square cavity with sidewall heating.

The deformation of a viscous drop moving under the action of gravity normal to a plane solid wall is studied. Under the assumption of creeping flow, the motion is studied as a function of the viscosity ratio between the drop and the suspending fluid, of surface tension, and of the initial drop configuration. Using the boundary integral formulation, the flow inside and outside the drop is represented in terms of a combined distribution of a single-layer and a double-layer potential of Green functions over the drop surface. The densities of these distributions are identified with the discontinuity in the interfacial surface stress, and with the interfacial velocity. The problem is formulated as a Fredholm integral equation of the second kind for the interfacial velocity which is solved by successive iterations. It is found that in the absence of surface tension, a drop moving away from the wall obtains an increasingly prolate shape, eventually ejecting a trailing tail. Depending on the initial drop configuration, ambient fluid may be entrained into the drop along or away from the axis of motion. Surface tension prevents the formation of the tail allowing the drop to maintain a compact shape throughout its evolution. The deformation of the drop has little effect on its speed of rise. A drop moving towards the wall obtains an increasingly oblate shape. In the absence of surface tension, the drop starts spreading in the radial direction reducing into a thinning layer of fluid. This layer is susceptible to the gravitational Rayleigh–Taylor instability. Surface tension restricts spreading, and allows the drop to attain a nearly steady hydrostatic shape. This is quite insensitive to the viscosity ratio and to the initial drop configuration. The evolution of the thin layer of fluid which is trapped between the drop and the wall is examined in detail, and with reference to film-drainage theory. It is shown that the assumptions underlying this theory are accurate when the surface tension is sufficiently large, and when the viscosity of the drop is of the same or lower order of magnitude as the viscosity of the ambient fluid. The numerical results are discussed with reference to film-drainage asymptotic theories.

Cavitation inception in a turbulent shear layer was studied at Reynolds numbers up to 2 × 106. Flash photography, high-speed motion pictures and holography were used to study the relation of cavitation inception to the shear-layer turbulent structure. Both spanwise and streamwise vortices were clearly visualized by the cavitation. Cavitation inception consistently occurred in the streamwise vortices and more fully developed cavitation was visible in both structures, with the streamwise cavities typically confined to the braid regions between adjacent spanwise vortices. The strength of the streamwise vortices was estimated using a Rankine vortex model, which showed that their strength was always less than 10% of that of the spanwise vortices. Measurements of fluctuating pressures were made by holographically monitoring the size of air bubbles injected into the non-cavitating shear flow. The measured pressure fluctuations had positive and negative peaks as high as 3 times the free-stream dynamic pressure, sufficient to explain cavitation inception at high values of the inception index. The occurrence of inception in the streamwise vortices of the shear layer, combined with previous reports of velocity dependence of the streamwise vortex strength, may explain the commonly observed Reynolds-number scaling of the cavitation inception index in shear flows.

When vortices are embedded in a shearing zonal flow their interactions are changed qualitatively. If the zonal flow's shear and the vortex's strength are of the same order and opposite sign, the vortex is pulled into a thin spiral, fragments, and is destroyed in a turn-around time. If the signs are the same, the vortex redistributes its vorticity so that its maximum value is at the centre, and its shape is determined by the ratio of its vorticity to the shear of the surrounding zonal flow. The dynamics depends crucially on the exchange between the self-energy of the vortices and the interaction energy of the zonal flow with the vortices. A numerical example that shows all of these effects is the breakup of a vortex layer: either a single large vortex is formed or successively smaller and more numerous thin filaments of vorticity are created. Two stable vortices are shown to merge if their initial separation in the cross-zonal direction is smaller than a critical distance which is approximately equal the vortices’ radii. The motions of large vortices are constrained by conservation laws, but when the zonal flow is filled with small-scale filaments of vorticity, the large vortices exchange energy with the filaments so that they are no longer constrained by these laws, and their dynamics become richer. Energy is shown to flow from the large vortices to the filaments, and this observation is used to predict the strength of boundary layers and the critical separation distance for vortex merging.

Shearing flow of an idealized, two-dimensional foam with monodisperse, spatially periodic cell structure is examined. Viscous effects are modelled by the film withdrawal mechanism of Mysels, Shinoda & Frankel. The primary flow occurs where thin films with inextensible interfaces are withdrawn from or recede into quasi-static Plateau borders, film junctions that contain most of the liquid. The viscous flow induces an excess tension that varies between films and alters the foam structure. The instantaneous structure and macroscopic stress for a foam of arbitrary orientation are determined for simple shearing and planar extensional flow. As the foam flows, the Plateau borders coalesce and separate, which leads to switching of bubble neighbours. The quasi-steady asymptotic analysis of the flow is valid for small capillary numbers Ca based on the macroscopic deformation rate. This requires the foam to be wet, i.e. the liquid volume fraction must be large enough that structure varies continuously with strain. The viscous contribution to the instantaneous stress is $O(Ca^{\frac{2}{3}})$ and depends on the foam orientation and liquid content. Viscometric functions are determined by time averaging the instantaneous stress. When these functions are scaled by surface tension over cell size, the shear stress is $O(Ca^{\frac{2}{3}})$; by contrast, the first normal stress difference is O(1). Even though wet foams are elastic for small but finite deformations, the time-averaged shear stress does not evidence a yield stress.

Exact expressions for the internal and external Froude numbers for two-layer flows are derived from the celerities of infinitesimal long internal and external waves, without recourse to the Boussinesq approximation. These expressions are functions of the relative density difference between the layers; the relative thickness of the layers; and the stability Froude number, which can be regarded as an inverse bulk Richardson number. A fourth Froude number, the composite Froude number, has been most often used in previous studies. However, the usefulness of the composite Froude number is shown to diminish as the stability Froude number increases. The potential confusion associated with having four Froude numbers of importance has been alleviated by deriving an equation interrelating them. This equation facilitates a comprehensive understanding of the hydraulics of two-layer flows.

It is demonstrated that in substantial portions of some flows (both Boussinesq and non-Boussinesq exchange flow through a contraction are presented as examples), the stability Froude number exceeds a critical value. In this case hydraulic analysis yields imaginary phase speeds corresponding to the instability of long internal waves. Various implications of this result are discussed.