The nonlinear evolution of instabilities of a plane-parallel shear flow with an inflection-point profile is studied. The particular example of the cubic-profile flow generated in an inclined layer heated from above and cooled from below is chosen because it exhibits supercritical bifurcations for secondary and tertiary flows. Since the limit of small Prandtl number is assumed, buoyancy effects caused by temperature perturbations are negligible. The analysis describes first the transition to transverse roll-like vortices which become unstable at slightly supercritical Grashof numbers to a vortex-pairing instability with alternating pairing in the spanwise direction. Three-dimensional finite-amplitude solutions for this tertiary mode of motion are computed and discussed. Finally the question of the stability of the tertiary flow is addressed.

We study the existence and the role of solitary waves in the instability of a fluid layer flowing down an inclined plane. The approach presented is fully nonlinear. Solitary waves steady in a moving frame are described by homoclinic trajectories of an associated ordinary differential equation. They are searched numerically for a given value of viscosity and surface tension. Several kinds of solitary waves can exist, characterized by their number n of humps. We investigate the stability of these waves by integrating the initial-value problem directly. Solitary waves with more than 1 hump did not appear in the simulation, and moreover a catastrophic behaviour took place for too large a Reynolds number (R [gsim ] R*1) or too large an amplitude, suggesting a finite-time singularity. The long-term evolution is shown to be a very slow relaxation to a steady state in a moving frame. The relation to the experimental observation of localized wavetrains is also discussed.

A rigid plane thin sheet is sliding steadily at speed U close to a plane wall, in a fluid of kinematic viscosity v. The sheet is infinitely wide and is of length L in the direction of motion, and its leading edge is a distance h0 [Lt ] L from the wall. A solution is sought for arbitrary finite values of R = Uh20/νL. In the limit as ε = h0/L→0, the problem reduces to that of solving the boundary-layer equation in the gap region between sheet and wall, and this is done here both by an empirical linearization, and by direct numerical methods. The solutions have the property that they reduce to those predicted by lubrication theory when R is small, and to those predicted by an inviscid small-gap theory when R is large. Special attention is paid to the correct entrance and exit conditions, and to the location of the centre of pressure.

One of the recent developments in harnessing wave energy is to construct a line of submerged structures parallel to the incident swell crests in order to transform the straight crests to circular crests converging to a focus. To understand the neighbourhood of the focus, we have carried out theoretical studies by accounting for diffraction and nonlinearity, both separately and jointly. Experiments have also been conducted in a large outdoor basin and are compared with the theories. These comparisons tend to favour the approximate nonlinear theory, but the efficiency of the focusing device as an energy concentrator does not appear to be significantly impaired by nonlinear effects.

Results of laboratory experiments are presented in which a finite volume of homogeneous fluid was released instantaneously into another fluid of slightly lower density. The experiments were performed in a channel of rectangular cross-section, and the two fluids used were salt water and fresh water. As previously reported, the resulting gravity current, if viscous effects are negligible, passes through two distinct phases: an initial adjustment phase, during which the initial conditions are important, and an eventual self-similar phase, in which the front speed decreases as t−1/3 (where t is the time measured from release). The experiments reported herein were designed to emphasize the inviscid motion. From our observations we argue that the current front moves steadily in the first phase, and that the transition to the inviscid self-similar phase occurs when a disturbance generated at the endwall (or plane of symmetry) overtakes the front. If the initial depth of the heavy fluid is equal to or slightly less than the total depth of the fluid in the channel, the disturbance has the appearance of an internal hydraulic drop. Otherwise, the disturbance is a long wave of depression. Measurements of the duration of the initial phase and of the speed and depth of the front during this phase are presented as functions of the ratio of the initial heavy fluid depth to the total fluid depth. These measurements are compared with numerical solutions of the shallow-water equations for a two-layer fluid.

A two-dimensional vortex pair is commonly generated by pushing fluid down a semi-infinite channel by means of an impulsively started piston. The strength and separation of the two fully developed vortices strongly depend upon the time history of the piston motion. When the piston is impulsively stopped, two secondary vortices are formed downstream of the channel ends and interact with the primary pair in a fairly complicated way.

In the present work we attempt to provide a discrete-vortex model of the process of pair formation. The effects of viscosity are assumed to affect only the separation process, having negligible influence on the overall flow. In the limit of infinite Reynolds number, the problem becomes one of inviscid flow, and the separation at the sharp edges is approximated by a Kutta–Joukowski condition, large vortex regions being replaced by simple concentrated vortices. The growing vortex sheets shed from the edge are represented by a simplified model due to Brown and Michael.

Present results are able to account for the failure of the ‘puffing’ technique as well as the success of Barker and Crow's ‘downwash’ technique in producing vortex pairs.

Flow-visualization experiments are also reported, and good qualitative agreement is found between numerical and experimental results.

The present model also shows that the presence of secondary vortices drastically modifies the trajectories of free vortices as obtained in a previous work due to Sheffield.

Two experiments (δ/R = 0.05 and 0.10) were performed to determine how boundary-layer turbulence is affected by strong convex curvature. The flow passed from a flat surface, over convex surface with 90° of turning, and then onto a flat recovery surface. The pressure gradient along the test surface was forced to be zero.

After the sudden introduction of curvature, the shear stress in the outer part of the boundary layer is sharply diminished. The wall shear also drops off quickly downstream. When the surface suddenly becomes flat again, the wall-shear and the shear-stress profiles recover very slowly towards flat-wall conditions. The shear-stress profiles in the curved region for both experiments collapse when $-\overline{uv}/u^2_{\tau}$ is plotted vs. distance from the wall normalized on wall radius, n/R. The strong-curvature data of So & Mellor also fall on the same curve. Thus suggests an asymptotic state for the shear-stress profiles of strongly curved boundary layers where R rather than boundary-layer thickness controls the active turbulence lengthscales. In this asymptotic region, the active shear-layer thickness is less than its initial value at the start of curvature. In the recovery region, the width of the active shear layer regrows slowly within the original velocity-gradient boundary layer, like a developing boundary layer under a free stream with a velocity gradient normal to the wall.

The shape of a vertical slender jet of fluid falling steadily under the force of gravity is studied. The problem is formulated as a nonlinear free boundary-value problem for the potential. Surface-tension effects are included and studied. The use of perturbation expansions results in a system of equations that can be solved by an efficient numerical procedure. Computations were made for jets issuing from three different orifice shapes, which were an ellipse, a square, and an equilateral triangle. Computational results are presented illustrating the effects of different values for the surfacetension coefficient on the shape of the jet and the periodic nature of the cross-sectional shapes.

A liquid, contained in a quarter plane, undergoes steady motion due to thermocapillary forcing on its upper boundary, a free surface separating the liquid from a passive gas. The rigid vertical sidewall has a strip whose temperature is elevated compared with the liquid at infinity. A boundary-layer analysis is performed that is valid for large Marangoni numbers M and Prandtl numbers P. It is found that the Nusselt number N for the horizontal heat transport satisfies $N \sim \min (M^{\frac{2}{7}}, M^{\frac{1}{5}} P^{\frac{1}{10}})$. Generalizations are discussed.

A circulatory flow called a ‘backwash vortex’ forms under waves running up a sloping bed. The backwash vortex is classified into two types: the ‘steady backwash vortex’ and the ‘unsteady backwash vortex’. The former is a steady streaming formed when Re2 < 90, and the latter an unsteady separated flow formed when Re2 > 90, where Re2 is a form of Reynolds number. The ratio of the lengthscale of the backwash vortex to the swash length is proportional to Re2−½ when Re2 [Lt ] 30, and becomes constant when Re2 is sufficiently large.

When a salinity gradient is heated at a single point, a hot plume is formed with a series of layers around and above it in a ‘Christmas-tree’ flow pattern. Qualitative visual observations of the development of this system are reported. Three kinds of layers are observed: the first kind is formed above the basic plume from a hierarchy of secondary plumes on top of the basic one, the second develops around the upper part of the basic plume, and is driven by the same mechanism as the layers in sidewall-heating experiments, and the third, forming around the lower part of the basic plume, is of the same nature as the second type but also has something in common with double-diffusive intrusions. A characteristic feature for all the three kinds of layers is the existence of systematic shearing motions and vortices. Quantitative estimates for the plume height and the thickness of the layers are obtained.

Numerical solutions of the Navier-Stokes equations for steady axisymmetric flow in the Taylor experiment are presented. In the cases considered the annulus is so short that only one or two Taylor cells are present. The results are compared with a recent theoretical and experimental study by Benjamin & Mullin (1981). The qualitative picture of the flows possible proposed by Benjamin & Mullin is confirmed by the calculations, and the quantitative agreement with their experimental results is quite satisfactory.

We describe experiments in which we have observed the onset of Rayleigh-Bénard convection in normal liquid 3He–4He mixtures. Evidence of overstability was seen when heating from below but only stationary convection was observed when heating from above. Measurements of the critical Rayleigh number are presented and compared with the predictions of current theories of marginal stability in a binary mixture. These experiments exemplify liquid 3He–4He mixtures as a system for the study of convective instabilities.

An experimental study of the stratified flow over triangular-shaped ridges of various aspect ratios is described. The flows were produced by towing inverted bodies through saline-water solutions with stable (linear) density gradients. Flow-visualization techniques were used extensively to obtain measurements of the lee-wave structure and its interaction with the near-wake recirculating region and to determine the height of the upstream dividing streamline (below which all fluid moved around, rather than over the body). The Froude number F(= U/Nh) and Reynolds number (Uh/ν), where U is the towing speed, N is the Brunt–Väisälä frequency, h is the body height, and ν is the kinematic viscosity, were in the nominal ranges 0.2–1.6 (and ∞) and 2000–16000 respectively. The study demonstrates that the wave amplitude can be maximized by ‘tuning’ the body shape to the lee-wave field, that in certain circumstances steady wave breaking can occur or multiple recirculation regions (rotors) can exist downstream of the body, that vortex shedding in horizontal planes is possible even at F = 0.3, and that the ratio of the cross-stream width of the body to its height has a negligible effect on the dividing streamline height. The results of the study are compared with those of previous theoretical and experimental studies where appropriate.

A numerical model based on a Lagrangian description has been developed for studying run-up of long water waves governed by a set of Boussinesq equations. The performance of the numerical scheme has been tested by comparing with analytical solutions and experimental data. Simulations of the run-up of solitary waves on relatively steep planes (inclination angle > 20°) show surface displacements and run-up heights in good agreement with experiments. For waves with relatively large amplitude the simulations reveal the development of a breaking bore during the backwash. Results for run-up heights in converging and diverging channels are also presented.

The properties of a fully three-dimensional surface gravity wave, the short-crested wave, are examined. Linearly, a short-crested wave is formed by two wavetrains of equal amplitudes and wavelengths propagating at an angle to each other. Resonant interactions between the fundamental and its harmonics are a major feature of short-crested waves and a major complication to the use at finite wave steepness of the derived perturbation expansion. Nonetheless, estimates are made of the maximum steepnesses, and wave properties are calculated over the range of steepnesses. Although results for values of the parameter θ near 20° remain uncertain, we find that short-crested waves can be up to 60% steeper than the two-dimensional progressive wave. At limits of the parameter range the results compare well with those for known two-dimensional progressive and standing water waves.

Fully three-dimensional surface gravity waves in deep water are investigated in the limit in which the length of the wave crests become long. We describe an analytic solution to fourth order in wave steepness, which matches onto known short-crested wave solutions on the one hand and onto the well-known two-dimensional progressivewave solution on the other. In the progressive-wave limit a particular solution in which the wave crests are semi-finite is given to sixth-order accuracy. These solutions are part of a more general set of solutions which are found from a nonlinear Schrödinger equation.

The mechanisms of displacement of one fluid by another are investigated in an etched network.

Experiments show that both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section. Calculation of displacement pressures are in good agreement with experiments for drainage, imbibition and removal of blobs. The results may be related to some flow behaviour exhibited in porous media.

Three intermolecular collision models have been used in Monte Carlo direct-simulation computations. Their merits have been assessed by comparing the predictions given for two contrasting flows with experimental results. In one flow viscous effects were predominant; in the other the rapid compression ahead of a blunt body was the feature concentrated upon. In both examples the flows were rarefied and hypersonic and the gas was diatomic and rotationally excited.