The two dominant, linearly independent surface-wave modes in a circular cylinder, which differ only by an azimuthal rotation of ½π and have equal natural frequencies, are nonlinearly coupled, both directly and through secondary modes. The corresponding, weakly nonlinear free oscillations are described by a pair of slowly modulated sinusoids, the amplitudes and phases of which are governed by a four-dimensional Hamiltonian system that is integrable by virtue of conservation of energy and angular momentum. The resulting solutions are harmonic in a particular, slowly rotating reference frame. Harmonic oscillations in the laboratory reference frame are realized for three special sets of initial conditions and correspond to a standing wave with a fixed nodal diameter and to two azimuthally rotating waves with opposite senses of rotation. The finite-amplitude corrections to the natural frequencies of these harmonic oscillations are calculated as functions of the aspect ratio d/a (depth/radius). A small neighbourhood of d/a = 0.1523, in which the natural frequency of the dominant axisymmetric mode approximates twice that of the two dominant antisymmetric modes, is excluded. Weak, linear damping is incorporated through a transformation that renders the evolution equations for the damped system isomorphic to those for the undamped system.

The weakly nonlinear, weakly damped response of the free surface of a liquid in a vertical circular cylinder that is subjected to a simple harmonic, horizontal translation is examined by extending the corresponding analysis for free oscillations. The problem is characterized by three parameters, α, β, and d/a, which measure damping, frequency offset (driving frequency–natural frequency), and depth/radius. The asymptotic (t↑∞) response may be any of: (i) harmonic (at the driving frequency) with a nodal line transverse to the plane of excitation (planar harmonic); (ii) harmonic with a rotating nodal line (non-planar harmonic); (iii) a periodically modulated sinusoid (limit cycle); (iv) a chaotically modulated sinusoid. It appears, from numerical integration of the evolution equations, that only motions of type (i) and (ii) are possible if 0.30 < d/a < 0.50, but that motions of type (iii) and (iv) are possible for all other d/a in some interval (or intervals) of β if α is sufficiently small. Only motion of type (i) is possible if α exceeds a critical value that depends on d/a.

The hydrodynamics and heat/mass transport associated with condensation on a moving drop have been investigated for the intermediate Reynolds-number range of drop motion (Re = O(100)). The drop environment is a mixture of saturated vapour and a non-condensable. The formulation entails a simultaneous solution of the quasi-steady elliptic partial differential equations that describe the flow field and transport in the gaseous phase, and the motion inside the liquid drop. The heat transport inside the drop is treated as a transient process. Results are reported for the interfacial velocities, drag, external and internal flow structure, heat flux, drop growth rate and temperature–time history inside the drop. The results obtained here have been compared with experimental data where available, and these show excellent agreement.

The results reveal several novel features. The surface-shear stress increases with condensation. The pressure level in the rear of the drop is higher. As a consequence, the friction drag is higher and the pressure drag is lower. The total drag coefficient increases with condensation rate for small values of drop size or temperature differential, and it decreases for large values of these parameters. The volume of the separated-flow region in the rear of the drop decreases with condensation. At very high rates of condensation, the recirculatory wake is completely suppressed. Condensation also delays the appearance of the weak secondary internal vortex motion in the drop. The heat and mass fluxes are significantly affected by the presence of the non-condensable in the gaseous phase and by the circulation inside the drop.

Potential flow past a porous body of arbitrary shape with constant physical permeability k0, as well as the flow in the corresponding porous medium, are analysed by means of a pair of linear Fredholm integral equations of the second kind. As an example for verification of the proposed general method, the case of a two-dimensional porous circular cylinder is worked out in detail.

The initial-value problem for Taylor columns is considered for the oceanographically relevant case of slow flow over obstacles of small slope, and horizontal scale of order of the fluid depth or larger. In such flows depth variations cause, in the neighbourhood of the obstacle, a gradient of background potential vorticity. Topographic Rossby waves, forced in the starting flow, propagate across the gradient, cycling the obstacle in times of order of the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the obstacle height. The linear inertialwave equation and the quasigeostrophic equations are related by considering the case where the inertial period is small compared with the topographic time, which is in turn small compared with the advection time.

When viscosity is present the waves decay to a steady motion in a time of order of the viscous spin-up time. In inviscid flow the waves do not decay. The flow oscillates about steady, irrotational, zero-circulation flow. The drag and lift on the obstacle oscillate with almost constant frequency and amplitude equal to the Coriolis force on a body of fluid whose volume matches that of the obstacle. Particles above a right cylinder move in closed orbits of diameter 1/S, where S is the topographic parameter introduced by Hide (1961). Particles away from the cylinder move irrotationally, but asymmetrically, from upstream to downstream. A shear layer is present at the boundary of the cylinder throughout the motion. The initial vorticity distribution for flow over a paraboloid is everywhere finite. However, regions of positive and negative vorticity interlace ever more closely during the motion, causing at large time a similar shear layer at the column boundary. By this time the vorticity is increasing without bound above the obstacle and neglected nonlinear, horizontal viscous, or vertical effects are important.

This paper reports an experimental study of a meandering water stream upon an inclined smooth hydrophobic surface. It was found that the sinuosity (ratio of the total stream length to the length of the projection of the stream on the line of maximum slope of the surface) increases with both increasing discharge rate and surface slope. It was observed that the meandering pattern is not always stable: once the discharge rate exceeds the upper critical value, the meandering pattern becomes unstable, whereas, when the discharge rate is smaller than the lower critical value, the water stream becomes discontinuous, and normally forms droplets, sliding successively down the sloping surface. It was found that, with increasing surface slope, the upper critical value decreases exponentially, while the lower critical value decreases only gradually. It was found that, when a system of stable meanders is formed on the surface, the meander loops are smoothly curved, swinging gradually from left-handed to right-handed deflections from the line of fastest descent, and vice versa, with an almost constant amplitude and wavelength. The stable meandering pattern migrates gradually down the sloping surface.

The observations showed that the central axis of the meandering stream does not coincide with the locus of the highest points of the stream, the highest points being displaced towards the outside of each bend: the cross-sectional profile of the stream is thus usually asymmetrical. It was found that the cross-sectional area of the stream varies cyclically, with one increase and one decrease associated with each bend of the stream. This cyclic variation is repeated many times along the length of the stream, with each point of maximum cross-sectional area located close to a bend. A secondary reversing spiral flow was observed in the stream, and it was found that the sense of rotation of the flow is reversed at each bend.

A plausible mechanism of these stream meanders is proposed on the basis of the present results, involving the existence of hysteresis of the contact angle between water and Plexiglas, the presence of asymmetrical surface-tension forces on the stream, and the acceleration and deceleration of the stream as it swings from loop to loop.

In this paper we investigate some effects of a boundary forcing on 2-dimensional cellular patterns in instabilities above threshold. Boundary forcing is modelled as an inhomogeneous boundary condition on the slowly varying amplitude A, i.e. $A = \lambda {\rm e}^{{\rm i}\phi_0}$ on boundaries. The relevant range is λ = O(ε½), where ε is the relative distance to the linear-instability threshold. A wavenumber-selection mechanism then occurs, resulting in a band of selected wavenumbers of width proportional to λ. For large values of λε−½ it is shown that no stationary solution exists outside the band of Eckhaus-stable wavenumbers (Eckhaus 1965). For finite geometries of size L, a nonlinear analogue of ‘quantization’ of modes is investigated. The amplitude equation (equivalent to a space-dependent Ginzburg–Landau model) is analysed by an expansion in powers of exp(-L/ζ), where ζ is the coherence length. The range λ = O(ε) is also investigated. A correction to previous theories of wavenumber selection through boundaries (Cross, Daniels, Hohenberg & Siggia 1983a; Pomeau & Zaleski 1981) is calculated. The latter results are general and assume only the existence of a higher-order stationary amplitude equation, which is recast in a form consistent with its boundary conditions.

This paper extends previous theoretical work on inextensible sails in two ways. First, the asymptotic expansion of sail camber in terms of the small angle of attack α is continued to include terms of order α3, because only at this order can one obtain information about the longitudinal static stability of a sail aerofoil. Secondly, to describe aerofoils such as those found in pterodactyl or bat wings, we also consider pretensioned membranes which acquire camber by stretching the surface material, or bending the supporting structure. Approximate formulae are derived for the aerodynamic coefficients in the limits of large and small tension, and the effect of sail and structural flexibility on these coefficients is discussed.

The two-dimensional steady-state turbulent-diffusion equation is solved by a separation-of-variables process that leads to a Sturm–Liouville eigenvalue problem. The general solution, for arbitrary velocity and diffusivity distributions, is shown to be in the form of an eigenfunction expansion. For a steady uniform flow in a wide, open channel the velocity distribution in the vertical is well approximated by a logarithmic law and the diffusivity distribution is approximately parabolic. For these distributions the power-series solution technique for ordinary differential equations is used to determine the eigenfunctions and eigenvalues. The solution is compared with the standard solution that Holley, Siemans & Abraham (1972) obtained for a uniform velocity and diffusivity distribution. Experimental results are presented, and these show that

(1) the use of the correct velocity and diffusivity distribution results in a significant improvement in the agreement between experiment and theory; and (2) close to the source the fluctuations of concentration are of the order of the mean values.

Recent experiments reveal that mean spreading rates of artificially excited shear layers undergo abrupt changes at certain discrete positions within the flow. This paper is concerned with the interaction between externally excited instability waves and these regions of relatively sudden change in mean flow. While the main emphasis is on the acoustic field produced by these interactions, the upstream influence at the nozzle lip is also considered.

The vortex loop observed in flow-visualization studies of boundary-layer transition has been investigated by mapping the instantaneous velocity and vorticity fields. All three velocity components have been measured with hot-film anemometers at numerous grid points in a measuring volume centred on the location where the vortex loop appears in flow-visualizaton studies. The instantaneous vorticity field has been computed from the velocity field, and the vortex loop is revealed in the longitudinal component of vorticity. The loop propagates downstream at approximately the primary disturbance wavespeed. The fluid in the outer part of the boundary layer travels faster, and flows over the loop. This forms the inflexional high-shear layer, which breaks down into the hairpin vortices. The magnitude of the vorticity in the high-shear layer is actually about three times larger than that in the loop. These two regions of vorticity are distinguished by the direction of the instantaneous vorticity vectors, i.e. the vectors in the high-shear layer run in the spanwise direction, while the vectors in the vortex loop run primarily in the downstream direction. This also explains why the loop cannot be detected with simple ∂u/∂y measurements.

Some new exact relations are derived between integral properties of a progressive irrotational gravity wave of finite amplitude in deep water. In particular it is shown that the Eulerian-mean angular momentum $\overline{A}_{\rm E}$ is directly proportional to the Lagrangian T−V, through the relation \[ \overline{A}_{\rm E} = 2c(T-V)/g, \] where c is the phase speed and g denotes the acceleration due to gravity. Moreover, for waves of constant length, the differential relation \[ {\rm d}\overline{A}_{\rm E} = 2(3T-V)\,{\rm d}c/g \] also holds.

In a wave of limiting steepness it was shown previously that the level of action ya is very nearly equal to the crest level ymax. This is further discussed, and is shown to be probably a numerical coincidence.

The transition from diffusion-dominated to inertia-dominated behaviour in the transport of condensable heavy molecules carried in a continuum subsonic He jet that impinges on a solid surface is studied experimentally. The Stokes number S, or ratio between the heavy-molecule relaxation time and the fluid-dynamic time, is varied in the interval 0 [lsim ] S [lsim ] 1 by changing the jet Mach number at a constant value of the Reynolds number. Although the heavy species departs considerably from equilibrium at all but the smallest values of S, the helium jet is always near equilibrium conditions. At values of S of order unity the observed rate of deposition at the stagnation point asymptotes to a value some six times greater than in the diffusion region (where S → 0), implying that the process is governed by the large inertia of the heavy species, very much like in aerosol impactors. As a result, it is argued that the concept of pressure diffusion is unsuitable to explain the observed behaviour. An approximate theoretical description of the transport process is given for the region S [Lt ] 1 where the kinetic problem is amenable to a hydrodynamic treatment. Finally, the analogy with the inertia-dominated behaviour of aerosols is used to assess the relative merits of various aerodynamics schemes aiming at separating isotopes.

We present experimental results on the modified Stokes force F exerted on a sphere in magnetic levitation whose position is kept fixed by an optical feedback system. A Newtonian liquid moves at a constant velocity U relative to the sphere. We consider the motion in two different situations.

(i) When the sphere approaches a wall perpendicular to U, the increase in |F| due to lubrication agrees quantitatively with theoretical results such as those of Brenner (1961) and Maude (1961), obtained neglecting the unsteadiness of the flow field.

(ii) In the complementary situation of a sphere moving along the axis of a cylindrical tube, our results expressed as a function of the eccentricity of the trajectory and of the ratio of the two radii confirm and extend previous theoretical analyses. They show in particular the existence of a minimum of |F| away from the axis of the cylinder and a sharp increase in |F| when the sphere approaches the sidewall. By comparing with the results for a sphere moving parallel to a flat wall, we analyse the effect of the curvature of the cylindrical tube.

There are two main effects of turbulence on the mean flow past rectangular cylinders, just as found earlier for square rods. Small-scale turbulence increases the growth rate of the separated shear layers through increased mixing. Large-scale turbulence weakens regular vortex shedding by reducing spanwise correlation. The consequences of increased mixing and weakened regular vortex shedding depend on the depth-to-height ratio of a rectangular cylinder. In particular, the mean base pressure of a rectangular cylinder can be varied significantly by both the scale and intensity of turbulence.

We consider the motion of a sphere or a slender body in the presence of a plane fluid–fluid interface with an arbitrary viscosity ratio, when the fluids undergo a linear undisturbed flow. First, the hydrodynamic relationships for the force and torque on the particle at rest in the undisturbed flow field are determined, using the method of reflections, from the spatial distribution of Stokeslets, rotlets and higher-order singularities in Stokes flow. These fundamental relationships are then applied, in combination with the corresponding solutions obtained in earlier publications for the translation and rotation through a quiescent fluid, to determine the motion of a neutrally buoyant particle freely suspended in the flow. The theory yields general trajectory equations for an arbitrary viscosity ratio which are in good agreement with both exact-solution results and experimental data for sphere motions near a rigid plane wall. Among the most interesting results for motion of slender bodies is the generalization of the Jeffrey orbit equations for linear simple shear flow.

The propagation of long, weakly nonlinear interfacial waves in a two-layer fluid of slowly varying depth is studied. The governing equations are formulated to include cubic nonlinearity, which dominates quadratic nonlinearity in some parametric neighbourhood of equal layer depths. Numerical solutions are obtained for an initial profile corresponding to either a single solitary wave or a rank-ordered pair of such waves incident in a monotonic transition between two regions of constant depth. The numerical solutions, supplemented by inverse-scattering theory, are used to investigate the change of polarity of the incident waves as they pass through a ‘turning point’ of approximately equal layer depths. Our results exhibit significant differences from those reported by Knickerbocker & Newell (1980), which were based on a model equation. In particular, we find that more than one wave of reversed polarity may emerge.

Turbulent spots formed artificially in a Blasius boundary layer have been investigated over a finite width extending across the plane of symmetry of the spots. Hot wires were used to measure the local mean and fluctuating parts of the downstream and spanwise velocity components, and the third component was computed. The results are presented in contour diagrams, and are compared with previous published work.

The values of energy thickness in the spots are computed from the contours of downstream velocity. The spots are found to consist of an upper part containing most of the turbulence, moving over a lower layer whose contribution to the energy thickness is small. When, near the rear of the spot, the energy thickness decreases to the value in Blasius flow, the two parts recombine, and the flow slowly regains the Blasius velocity profile.

The spots grow by entrainment of laminar fluid, and the physical principles governing this process are discussed.

A class of compressible laminar boundary-layer flows subject to adverse pressure gradients of different magnitude is studied using a finite-element–differential method in which the assumed solutions are represented by classical cubic spline functions. The numerical integration process for the reduced initial-value problem has been carried out directly to at least one integration step upstream of the separation point, and very accurate numerical results have been obtained for a large number of integration steps extremely close to separation. The skin-friction and heat-transfer coefficients for nearly zero-heat-transfer, cooled-wall and heated-wall cases, computed under the assumption of constant Prandtl number Pr = 1.0 as well as Pr = 0.72, have clearly exhibited the same distinctive behaviour near separation. It is deduced that Buckmaster's series expansions for the solution near separation, derived on the assumptions of cooled wall and Pr = 1.0, are valid for all the cases considered. By matching the numerical results with Buckmaster's expansions, accurate distributions of skin friction and heat transfer have been obtained up to the separation point. Moreover, the importance of Prandtl number on the solution is evidenced from the numerical results presented.

All three velocity fluctuations and the temperature fluctuation have been measured in a slightly heated turbulent plane jet. Attention is focused on the interaction region of the flow, which is situated between the location where the two mixing layers nominally merge and that which corresponds to approximate self-preservation.

For the jet considered here the mixing-layer structures are symmetrical with respect to the centreline, and when they meet in the interaction region the redistribution of turbulence quantities is dramatic. This redistribution is examined in detail. Also examined is the effect of the generation, in the interaction region, of new structures, asymmetric with respect to the centreline, which evolve into the self-preserving flow region downstream.

Turbulence parameters, such as the turbulent Prandtl number, probability density functions, skewness and flatness factors, are also presented, primarily to guide computer simulations of this flow. The superposition procedure of Weir, Wood & Bradshaw (1981), which assumes that the turbulence structure of each mixing layer is not significantly altered by the interaction, is not appropriate to the present flow.