The effects of surface tension σ and tube inclination β on the Froude number Fr of a large bubble rising in a two-dimensional duct is considered. It is found that there exists either one (for small σ and β > 0°) or a set (for any σ and β = 0°) of Fr-values for which the bubble has a continuous derivative at the nose. By selecting either this single Fr (or the maximum of the set), we obtain solutions in excellent agreement with both theoretical predictions and experimental results.

Water waves for which both gravitation and surface tension are important (ripples) exhibit a variety of instabilities. Here, experimental results are presented for ripple wavetrains on deep water with frequencies greater than 19.6 Hz where a continuum of resonant triad interactions are dynamically admissible. The experimental wave-trains are indeed unstable, and the instability becomes more pronounced as non-linearity is increased. The unstable wavefield is characterized by significant spatial disorder while temporal measurements at fixed spatial locations remain quite ordered. In fact, for most experiments temporal measurements suggest that a selection process exists in which a single triad dominates evolution. The dominant triad typically does not involve a subharmonic frequency of the generated wave and persists over a wide range of amplitudes for the initial wave. Viscosity does not appear to be important in the selection process; however, it may be responsible for the lack of subsequent triad production by the excited waves of the initial triad. The presence of a selection process contradicts previous conjecture, based on the form of the interaction coefficients, that a broad-banded spectrum of waves should occur. The general absence of subharmonic growth also contradicts previously reported experiments. Results are also presented for wavetrains at the parametric boundary of 19.6 Hz and a degenerate case of resonant triads at 9.8 Hz (Wilton's ripples). In addition to resonant triads, the experiments show evidence of (generally) weaker narrow-band interactions.

The first part of this paper (§2) briefly reviews the history of the idea of the resonant nature of laminar-boundary-layer breakdown. In the second part a new wave-resonance concept of the breakdown mechanism is proposed. The existing experimental data on the laminar boundary layer (and plane channel flow) breakdown are analysed from the viewpoint of this concept and are compared with the well-known local high-frequency secondary-instability concept. The results testify to the correctness of the proposed wave-resonant concept for the initial stages of breakdown, in particular for the K-regime of transition up to the spike formation and its doubling.

Within the framework of the wave-resonance concept, before constructing the corresponding theory, many important features of the disturbance development can be qualitatively explained and understood. Concerning the understanding of the spike appearance, the wave-resonance concept complements the local high-frequency secondary-instability one and represents by itself a new fruitful viewpoint on this phenomenon. The development of the wave-resonance concept and its application to the analysis of numerical and physical experiments, together with the construction on this basis of the corresponding theory, can give an essential impetus towards the better understanding of the breakdown nature.

This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg-de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon.

To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, Ts, and the scaled amplitude α of the solitons so generated are related by the formula Ts = const α−3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.

This paper concerns the conversion of momentum and energy from evanescent surface waves into sound. Exact results are obtained from surface waves of specified form on a confined region of an otherwise rigid plane surface. The model chosen is simple enough for exact analysis while approximating some of what we believe to be significant aspects of sound generation by vibrating surface panels.

We find that the evanescent wave approaching an edge gives up all of its energy into sound, a sound which is beamed mainly parallel to the direction of the surface-wave phase velocity. The surface remains energetically inactive, but exerts a force on the fluid in the opposite direction to the incoming wave. This force is balanced by a nonlinear mean pressure gradient in the field of the evanescent wave, and by momentum in the sound field.

Sound is also produced when a similar evanescent wave emerges from an edge. The surface has then to provide the necessary energy for both waves. These waves induce a mean axial force at the boundary which forces the fluid in the direction of the receding evanescent wave.

A similar wave travelling across a finite panel in the otherwise rigid plane surface is observed to have some characteristics of the previous two cases, but there is no axial force arising from the mean pressure gradient.

These results are applied to the problem of a semi-infinite tensioned membrane, and the energy radiation under light fluid loading is determined for the case of high and low free membrane wave speeds.

Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptotic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices’ deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighbouring vortices.

The natural convection from a heated circular cylinder in an unbounded region of porous medium is investigated for the full range of Rayleigh numbers. At small Rayleigh numbers a qualitative solution is obtained and at large Rayleigh numbers the second-order boundary-layer solution is found that takes into account the first-order plume solution. In order to find the solution at finite Rayleigh numbers the two governing coupled, nonlinear, elliptic partial differential equations are expressed in finite-difference form using a specialized technique which is second-order accurate everywhere. Further, methods are devised which deal with the plume and infinity boundary conditions. Although numerical results are presented for Rayleigh numbers up to 400 solutions of the finite-difference equations can be obtained for higher values of the Rayleigh numbers but in these cases the mesh size used is too large to adequately deal with the developing boundary-layer on the cylinder and the plume.

The numerical results show how the theories at both low and high Rayleigh numbers are approached. The plume solution which develops with increasing Rayleigh number agrees with that predicted by the theory presented using the boundary-layer approximation. No separation of the flow at the top of the cylinder is found and there are no indications that it will appear at higher values of the Rayleigh number. The results presented here give reasonable agreement with the existing experimental results for Rayleigh numbers of order unity. However as the Rayleigh number increases to order 102 there is a large discrepancy between the theoretical and experimental results and this is because at these higher values of the Rayleigh number the Darcy approximation has been violated in the experimental results. This indicates the severe limitations of some of the existing theories which use boundary-layer analyses and the Darcy approximation for flows in a porous medium. The application of Darcy's law requires that the size of the pores be much smaller than the scale of the bulk flow and inertial and thermal lengthscales.

A weakly nonlinear model is developed from the Hamiltonian formulation of water waves, to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, non-symmetric Wilton's ripples exist. They appear via a spontaneous symmetrybreaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and that further non-symmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low-gravity experiments is postulated.

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

Laminar-film flow with condensation or evaporation on a vertical fluted cylinder is examined. A kinematic wave equation describing the evolution of the film profile is obtained and solutions presented. The film profile evolves owing to axial, gravity-driven flow and transverse, surface-tension-driven flow from the crests to the valleys of the fluted cylinder. In the case of condensation the majority of the film reaches a uniform thickness and consequently there is a significant improvement in heat transfer compared with the classical unfluted result where the film thickens in an unbounded fashion. For evaporation a critical value of a parameter which involves the ratio of the Weber number and the gradient of the surface curvature is found below which the film totally drys out and above which the fluid funnels into tapering rivulets and only partially drys out. Typical dry-out lines are presented. For short cylinders the evaporative mass transfer for a fluted cylinder is slightly greater than that predicted for an unfluted case. However, when the cylinder is long the mass transfer is far less for a fluted cylinder owing to the reduction in film area associated with partial film dry out.

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number (N = O(1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness (ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2√2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Steady and unsteady velocity components over a backward-facing circular arc are measured by laser-Doppler velocimetry. A periodic disturbance is added to the mean flow and the response of unsteady separation is investigated. Special attention is given to the distribution and the flux of vorticity.

Two approaches have been used to study the torsion effect on the fully developed laminar flow in a helical pipe of constant circular cross-section. The first approach is the series expansion method that perturbs the Poiseuille flow and is valid for low Dean numbers with both the dimensionless curvature and dimensionless torsion being much less than unity. The second is a numerical procedure that solves the complete Navier-Stokes equation and is applicable to intermediate values of the Dean number. The results obtained indicate that, as far as the secondary flow patterns are concerned, the presence of torsion can produce a large effect if the ratio of the curvature to the torsion is of order unity. In these cases the secondary flow, though still consisting of a pair of vortices, can be very much distorted. Under extreme conditions one vortex is so prevalent as to squeeze the second one into a narrow region. However, ordinarily the torsion effect is small and the secondary flow has the usual pattern of a pair of counter-rotating vortices of nearly equal strength. Concerning the flow resistance in the pipe the effect of torsion is always small in all the circumstances that have so far been considered.

We consider the axisymmetric thin liquid film formed on a horizontally spinning disk. The asymptotic structure of the steady film is obtained, after which a theory is developed to describe the evolution of localized disturbances imposed upon the steady film. It is shown that this can lead to the propagation of large gradients in the film. Moreover, it is found that under certain conditions the steady film can become unstable.

The rectified flow, induced by divergence of the vorticity flux in lateral oscillatory viscous boundary layers along the sidewalls of a semi-enclosed basin, is studied as a function of the Strouhal number, k, equivalent to the Reynolds number of the viscous inner oscillatory boundary layer, and of the Stokes number. The squared ratio of these numbers defines another Reynolds number, measuring the strength of the self-advection by the residual flow. For strong self-advection the residual current decays to zero in an outer boundary, its width being large compared to the width of the inner layer. The regimes of small, moderate and strong self-advection are analysed.

The upper surface of a large gas bubble rising steadily through liquid under gravity is a statically unstable interface, and if the liquid were stationary small sinusoidal disturbances to the interface with wavelength exceeding the critical value Λc determined by surface tension would grow exponentially. The existence of the deforming motion of the liquid adjoining the interface of a steadily rising bubble changes the nature of the problem of stability. It is shown that a small sinusoidal disturbance of the part of the interface that is approximately plane and horizontal remains sinusoidal, although with exponentially increasing wavelength. The amplitude of such a disturbance increases, from the instant at which Λ = Λc until Λ becomes comparable with the radius of curvature of the interface (R), and the largest amplification occurs for a disturbance whose initial wavelength is approximately equal to Λc. With a plausible guess at the disturbance amplitude and wavelength at which bubble break-up due to nonlinear effects is inevitable, it is possible to obtain an approximate numerical relation between the initial magnitude of the disturbance and the maximum value of R for which a bubble remains intact. This relation applies both to a spherical-cap bubble in a large tank and a bubble rising in a vertical tube in which the liquid far ahead of the bubble is stationary. The few published observations of the maximum size of spherical-cap bubbles are not incompatible with the theory, but lack of information about the magnitude of the ambient disturbances in the liquid precludes any close comparison.

The topology of the organized motion has been obtained in the slightly heated self-preserving far wake of a circular cylinder at a Reynolds number, based on the cylinder diameter, of about 1200. In a frame of reference moving with the organized motion, the toplogy in the plane of main shear reduces to a succession of centres and saddles, located at about the wake half-width. Centres are identifiable by large values of spanwise vorticity associated with the coherent large-scale motion. Saddles occur at the intersection of converging and diverging separatrices, the latter being identifiable with the high strain rate due to the large-scale motion. Large values of the longitudinal turbulence intensity associated with the smaller-scale motion occur at the centres. High values of the normal and shear stresses, the temperature variance and heat fluxes associated with the large-scale motion occur on either side of each saddle point along the direction of the diverging separatrix. Contours for the production of energy and temperature variance associated with the small-scale motion are aligned along the diverging separatrices, and have maxima near the saddle point. Contours for one component of the dissipation of small-scale temperature variance also have a high concentration along the diverging separatrix. Flow visualizations in the far wake suggest the existence of groups of three-dimensional bulges which are made up of clusters of vortex loops.

Data which describe the unidirectional spreading of several pure oils and oilsurfactant mixtures on water in the surface-tension regime are reported. Leadingedge position and profiles of velocity, thickness and film tension are given as functions of time. The data are consistent with the numerical similarity solution of Foda & Cox (1980), although the measured dependence of the film tension on the film thickness often differs from the equilibrium relationship. The configuration of the oil film near the spreading origin may be either a coherent multimolecular layer or a multitude of thinning, outward-moving lenses surrounded by monolayer. The pure oils show an acceleration zone connecting the slow-moving inner region to a fast-moving outer region, while the oil-surfactant mixtures show a much more gradual increase in film velocity.

The interaction of eddies with variations in topography, together with a tendency for large-scale wave propagation, generates a systematic stress which acts upon large-scale mean flows. This stress resists the midlatitude tropospheric westerlies, resists the oceanic Antarctic Circumpolar Current, and may be a dominant mechanism in driving coastal undercurrents. Associated secondary circulation provides a systematic upwelling in coastal oceans, pumping deeper water onto continental shelf areas. The derivation rests in turbulence closure theory and is supported by numerical experiments.