An investigation was undertaken to improve our understanding of low-Reynolds-number turbulent boundary layers flowing over a smooth flat surface in nominally zero pressure gradients. In practice, such flows generally occur in close proximity to a tripping device and, though it was known that the flows are affected by the actual low value of the Reynolds number, it was realized that they may also be affected by the type of tripping device used and variations in free-stream velocity for a given device. Consequently, the experimental programme was devised to investigate systematically the effects of each of these three factors independently. Three different types of device were chosen: a wire, distributed grit and cylindrical pins. Mean-flow, broadband-turbulence and spectral measurements were taken, mostly for values of Rθ varying between about 715 and about 2810. It was found that the mean-flow and broadband-turbulence data showed variations with Rθ, as expected. Spectra were plotted using scaling given by Perry, Henbest & Chong (1986) and were compared with their models which were developed for high-Reynolds-number flows. For the turbulent wall region, spectra showed reasonably good agreement with their model. For the fully turbulent region, spectra did show some appreciable deviations from their model, owing to low-Reynolds-number effects. Mean-flow profiles, broadband-turbulence profiles and spectra were found to be affected very little by the type of device used for Rθ ≈ 1020 and above, indicating an absence of dependence on flow history for this Rθ range. These types of measurements were also compared at both Rθ ≈ 1020 and Rθ ≈ 2175 to see if they were dependent on how Rθ was formed (i.e. the combination of velocity and momentum thickness used to determine Rθ). There were noticeable differences for Rθ ≈ 1020, but these differences were only convincing for the pins, and there was a general overall improvement in agreement for Rθ ≈ 2175.

The effects of compressibility on free (unsteady) viscous heat-conducting vortices are investigated. Analytical solutions are found in the limit of large, but finite, Reynolds number, and small, but finite, Mach number. The analysis shows that the spreading of the vortex causes a radial flow. This flow is given by the solution of an ordinary differential equation (valid for any Mach number), which gives the dependence of the radial velocity on the tangential velocity, density, and temperature profiles of the vortex; estimates of the radial velocity found by solving this equation are found to be in good agreement with numerical solutions of the full equations. The experiments of Mandella (1987) also report a radial flow in the vortex, but their estimates are much larger than the analytical predictions, and it is found that the flow inferred from the iexperiments violates the Second Law of Thermodynamics for two-dimensional axisymmetric flow. It is speculated that three-dimensionality is the cause of this discrepancy. To obtain detailed analytical solutions, the equations for the viscous evolution are expanded in powers of Mach number, M. Solutions valid to O(M2), are discussed for vortices with finite circulation. Two specific initial conditions – vortices with initially uniform entropy and with initially uniform density – are analysed in detail. It is shown that swirling axisymmetric compressible flows generate negative radial velocities far from the vortex core owing to viscous effects, regardless of the initial distributions of vorticity, density and entropy.

We generalize the class of models of the wall layer of Aubry et al. (1988), based on the proper orthogonal decomposition, to permit uncoupled evolution of streamwise and cross-stream disturbances. Since the Reynolds stress is no longer constrained, in the absence of streamwise spatial variations all perturbation velocity components eventually decay to zero. However, their transient behaviour is dominated by ’ghosts’ of the non-trivial fixed points and attracting heteroclinic cycles which are characteristic features of those models based on empirical eigenfunctions whose individual velocity components are fixed. This suggests that the intermittent events observed in Aubry et al. do not arise solely because of the effective closure assumption incorporated in those models, but are rooted deeper in the dynamical phenomenon of the wall region.

Over the past decade and a half, analyses of the dynamics of fluids containing moving contact lines have specified hydrodynamic models of the fluids in a rather small region surrounding the contact lines (referred to as the inner region) which necessarily differ from the usual model. If this were not done, a singularity would have arisen, making it impossible to satisfy the contact-angle boundary condition, a condition that can be important for determining the shape of the fluid interface of the entire body of fluid (the outer region). Unfortunately, the nature of the fluids within the inner region under dynamic conditions has not received appreciable experimental attention. Consequently, the validity of these novel models has yet to be tested.

The objective of this experimental investigation is to determine the validity of the expression appearing in the literature for the slope of the fluid interface in the region of overlap between the inner and outer regions, for small capillary number. This in part involves the experimental determination of a constant traditionally evaluated by matching the solutions in the inner and outer regions. Establishing the correctness of this expression would justify its use as a boundary condition for the shape of the fluid interface in the outer region, thus eliminating the need to analyse the dynamics of the fluid in the inner region.

Our experiments consisted of immersing a glass tube, tilted at an angle to the horizontal, at a constant speed, into a bath of silicone oil. The slope of the air–silicone oil interface was measured at distances from the contact line ranging between O.O13a. and O.17a, where a denotes the capillary length, the lengthscale of the outer region (1511 μm). Experiments were performed at speeds corresponding to capillary numbers ranging between 2.8 × 10-4 and 8.3 × 10-3. Good agreement is achieved between theory and experiment, with a systematic deviation appearing only at the highest speed. The latter may be a consequence of the inadequacy of the theory at that value of the capillary number.

A layer of volatile viscous liquid drains down a uniformly heated inclined plate. Long-wave instabilities of the uniform film are studied by deriving an evolution equation for two-dimensional disturbances. This equation incorporates viscosity, gravity, surface tension, thermocapillarity, and evaporation eifects. The linear theory derived from this describes the competition among the instabilities. Numerical solution of the evolution equation describes the finite-amplitude behaviour that determines the propensity for dryout of the film. Among the phenomena that appear are the tendency to wave breaking, the creation of secondary structures, and the preemption of dryout by mean flow.

Generalized Taylor dispersion theory is extended so as to enable the analysis of the transport in unbounded homogeneous shear flows of Brownian particles possessing internal degrees of freedom (e.g. rigid non-spherical particles possessing orientational degrees of freedom, flexible particles possessing conformational degrees of freedom, etc.). Taylor dispersion phenomena originate from the coupling between the dependence of the translational velocity of such particles in physical space upon the internal variables and the stochastic sampling of the internal space resulting from the internal diffusion process.

Employing a codeformational reference frame (i.e. one deforming with the sheared fluid) and assuming that the eigenvalues of the (constant) velocity gradient are purely imaginary, we establish the existence of a coarse-grained, purely physical-space description of the more detailed physical-internal space (microscale) transport process. This macroscale description takes the form of a convective–diffusive ‘model’ problem occurring exclusively in physical space, one whose formulation and solution are independent of the internal (‘local’-space) degrees of freedom.

An Einstein-type diffusion relation is obtained for the long-time limit of the temporal rate of change of the mean-square particle displacement in physical space. Despite the nonlinear (in time) asymptotic behaviour of this displacement, its Oldroyd time derivative (which is the appropriate one in the codeformational view adopted) tends to a constant, time-independent limit which is independent of the initial internal coordinates of the Brownian particle at zero time.

The dyadic dispersion-like coefficient representing this asymptotic limit is, in general, not a positive-definite quantity. This apparently paradoxical behaviour arises due to the failure of the growth in particle spread to be monotonic with time as a consequence of the coupling between the Taylor dispersion mechanism and the shear field. As such, a redefinition of the solute's dispersivity dyadic (appearing as a phenomenological coefficient in the coarse-grained model constitutive equation) is proposed. This definition provides additional insight into its physical (Lagrangian) significance as well as rendering this dyadic coefficient positive-definite, thus ensuring that solutions of the convective–diffusive model problem are well behaved. No restrictions are imposed upon the magnitude of the rotary Péclet number, which represents the relative intensities of the respective shear and diffusive effects upon which the solute dispersivity and mean particle sedimentation velocity both depend.

The results of the general theory are illustrated by the (relatively) elementary problem of the sedimentation in a homogeneous unbounded shear field of a size-fluctuating porous Brownian sphere (which body serves to model the behaviour of a macromolecular coil). It is demonstrated that the well-known case of the translational diffusion in a homogeneous shear flow of a rigid, non- fluctuating sphere (for which the Taylor mechanism is absent) is a particular case thereof.

The events which are responsible for strong Reynolds-stress production in the near-wall region of a bounded turbulent shear flow have been investigated in a turbulent boundary layer at a Reynolds number based on momentum thickness of Reθ = 4650. The coherent structures associated with the production process have been studied using the quadrant detection technique. All three velocity components were measured in a three-dimensional sampling volume about the point of detection. The conditional ensemble-averaged velocity field associated with the detection of a sweep or an ejection is presented and compared with non-conditioned space–time correlations. Conditional space–time probability density distributions were calculated at all measurement locations based on the occurrence of a Reynolds-stress-producing event at the detection point. The resulting three-dimensional representation of the conditional probability demonstrates that a significant fraction of the events are relatively large in scale, that a hierarchy of sizes exists and that there is a link between the outer flow and the ’bursting’ process. However, many investigators have shown that the ’bursting’ frequency scales with wall variables. Therefore all indications suggest that the scales are generated by a wall-layer mechanism but grow to sizes and convect with velocities scaling with the outer layer.

A complete second-order solution is presented for the two-dimensional wave motion forced by a generic planar wavemaker. The wavemaker is doubly articulated and includes both piston and hinged wavemakers of variable draught. It is shown that the first-order evanescent eigenseries cannot be neglected when computing the amplitude of the second-order free wave. A previously neglected, time-independent solution that is required to satisfy an inhomogeneous kinematic boundary condition on the wavemaker as well as an mhomogeneous Neumann boundary condition on the free surface is examined in detail for the first time. This time-independent solution is found to accurately estimate the mean return flow in a closed wave flume computed by the Eulerian method. This mean return current due to Stokes drift is usually estimated using the principle of kinematic conservation of mass flux. Even though the first-order eigenseries will converge for any geometry of a generic planar wavemaker, the second-order solutions obtained from Stokes perturbation expansions will not converge for all planar wavemaker geometries.

This article considers certain two-dimensional, irrotational, steady flows in fluid regions of finite depth and infinite horizontal extent. Geometrical information about these flows and their singularities is obtained, using a variant of a classical comparison principle. The results are applied to three types of problems: (i) supercritical solitary waves carrying planing surfaces or surfboards, (ii) supercritical flows past ship hulls and (iii) supercritical interfacial solitary waves in systems consisting of two immiscible fluids.

Disturbed flow over a rotating disk can lead to transition of the von Kármán boundary layer at a much lower Reynolds number, Re, (i.e. smaller radius) than that due to the well-known Type 1 stationary mode of instability. This early transition is due to the excitation of the Type 2 instability, similar to that found in the Ekman layer. Detailed numerical values of the growth rates, phase speeds, group velocities, neutral curves, and other characteristics of these two instabilities have been calculated over a wide range of parameters. Neutral curves for the Ekman and Bödewadt boundary layers also are presented. The minimum critical Reynolds numbers for the von Kármán, Ekman and Bödewadt layers are Rec(2) = 69.4, 54.3, and 15.1 with wavelengths L = 22.5, 20.1, and 16.6 and at angles ε = −19.0°, −23.1°, and −33.2°, respectively. These minimum critical values frequently do not well describe laboratory observations, however, because at larger Re other modes grow more rapidly and dominate the flow.

The computed results are in excellent agreement with laboratory observations wherever comparison is possible. The growth of representative Type 1 instabilities with radius is shown to lead to N-factors greater than 9 at Re = 520 as appears to be necessary for transition to turbulence by the interaction of Type 1 with the basic flow. The growth of Type 2 instabilities with radius can lead to three additional mechanisms of transition. The necessary levels of excitation of Type 2 for these different mechanisms are estimated.

A sequence of photographs from a ciné film illustrate one of the transition mechanisms discussed: the interaction of Type 2 instabilities and a secondary instability that is nearly perpendicular to the Type 2 vortices.

We study the inviscid mechanisms governing the three-dimensional evolution of an axisymmetric jet by means of vortex filament simulations. The spatially periodic calculations provide a detailed picture of the processes leading to the concentration, reorientation, and stretching of the vorticity. In the purely axisymmetric case, a wavy perturbation in the streamwise direction leads to the formation of vortex rings connected by braid regions, which become depleted of vorticity. The curvature of the jet shear layer leads to a loss of symmetry as compared to a plane shear layer, and the position of the free stagnation point forming in the braid region is shifted towards the jet axis. As a result, the upstream neighbourhood of a vortex ring is depleted of vorticity at a faster rate than the downstream side. When the jet is also subjected to a sinusoidal perturbation in the azimuthal direction, it develops regions of counter-rotating streamwise vorticity, whose sign is determined by a competition between global and local induction effects. In a way very similar to plane shear layers, the streamwise braid vorticity collapses into counter-rotating round vortex tubes under the influence of the extensional strain. In addition, the cores of the vortex rings develop a wavy dislocation. As expected, the vortex ring evolution depends on the ratio R/θ of the jet radius and the jet shear-layer thickness. When forced with a certain azimuthal wavenumber, a jet corresponding to R/θ = 22.6 develops vortex rings that slowly rotate around their unperturbed centreline, thus preventing a vortex ring instability from growing. For R/θ = 11.3, on the other hand, we observe an exponentially growing ring waviness, indicating a vortex ring instability. Comparison with stability theory yields poor agreement for the wavenumber, but better agreement for the growth rate. The growth of the momentum thickness is much more dramatic in the second case. Furthermore, it is found that the rate at which streamwise vorticity develops is strongly affected by the ratio of the streamwise and azimuthal perturbation amplitudes.

The vortices near the origin of an initially laminar mixing layer have a single frequency with a well-defined phase; i.e. there is little phase jitter. Further downstream, however, the phase jitter increases suddenly. Even when the flow is forced, this same transition is observed. The forcing partially loses its influence because of the decorrelation of the phase between the forcing signal and the passing coherent structures. In the present investigation, this phenomenon is documented and the physical mechanism responsible for the phase decorrelation is identified.

The dependence of the parameters of capillary–gravity ripples on the characteristics of the steep surface waves (in the range 4–20 cm) that excite them is found. For steep 4–6 cm waves calculations are performed on the basis of the improved first Stokes method. Qualitative coincidence of the theoretical results with the experimental data is shown. For 7–20 cm waves the results are obtained by the multiple-scale method where the large-scale motion and the driving force for the ripple are found by the improved first Stokes method. Qualitative agreement between theory and experiment in this wavelength range is achieved.

For the free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, analyzed previously, the shape evolution was described in terms of a time-dependent mapping function z = Ω(ζ,t) of the unit circle, conformal on |ζ| [les ] 1. An equation giving the time evolution of the map, typically in parametric form, was derived. In this article, the flow of the infinite region exterior to a hypotrochoid is given. This includes the elliptic hole, which shrinks at a constant rate with a constant aspect ratio. The theory is extended to a class of semi-infinite regions, mapped from Im ζ [les ] 0, and used to solve the flow in a half-space bounded by a certain groove. The depth of the groove ultimately decays inversely with time.

Air bubbles released from an underwater nozzle emit an acoustical pulse which is of interest both for the study of bubble detachment and for elucidating the mechanism of sound generation by a newly formed bubble. In this paper we calculate theoretically the sequence of bubble shapes from a given nozzle and show that there is for each nozzle a bubble of maximum volume vmax Assuming that the bubble becomes detached at its ‘neck’, and that the volume of the detached bubble equals the volume V* of the undetached bubble above its ’neck’, we determine for each nozzle diameter D an acoustic frequency f* corresponding to 'slow’ bubble release.

Experiments show that the acoustic frequency, hence the bubble size, depends on the rate of air.flow to the bubble, but for slow rates of flow the frequency f is very close to the theoretical frequency f*.

High-speed photographs suggest that when the bubble pinches off. the limiting form of the surface is almost a cone. This is accounted for by assuming a line sink along the axis of symmetry. Immediately following pinch-off there is evidence of the formation of an axial jet going upwards into the bubble. This may play a part in stimulating the emission of sound.

Results of two field experiments in the North Sea are presented. Pressure was measured at two fixed heights above the mean water level and correlated with simultaneous wave height measurements. Roughly 90 hours of data have been analysed and the results are in agreement with earlier results obtained by Snyder et al. (1981). Measurements over swell give no indication of wave decay or growth for waves travelling faster than the wind or against the wind.

It is well known that the spectral peak of wind-induced gravity waves on the sea surface tends to shift to lower frequencies as the fetch increases. In past theories the nonlinear dynamics subsequent to Benjamin–Feir instability has been found to initiate the downshift in narrow-banded waves in the absence of wind. However, these weakly nonlinear theories all predict the downshift to be only the first phase of an almost cyclic process. Limited by the length of a wave tank, existing experiments are usually made with relatively steep waves which often break. Although there is a theory on how breaking adds dissipation to stop the reversal of the initial trend of downshift, the details of breaking must be crudely characterized by semi-empirical hypotheses.

Since the direct role of wind itself must be relevant to the entire development of wind-wave spectrum, we examine here the effect of wind on the nonlinear evolution of unstable sidebands in narrow-banded waves. We assume that the waves do not break and consider the case where the nonlinear effects that initiate the downshift, energy input by wind and damping by internal dissipation all occur on the same timescale. This means that not only must the waves be mild but the wind stress intensity must also lie within a certain narrow range. With these limitations we couple the air flow above the waves with Dysthe's extension of the cubic Schrödinger equation, and examine the initial as well as the long-time evolution of a mechanically generated wavetrain. For a variety of wind intensities, downshift is indeed found to be enhanced and rendered long lasting.

A dilute dispersion containing drops of one fluid dispersed in a second, immiscible fluid is considered. The drops are sufficiently small that inertia is negligible and that they remain spherical. Two drops of different size are in relative motion due to either Brownian diffusion or gravitational sedimentation. When the drops become close, they interact with each other owing to hydrodynamic disturbances and van der Waals attractions, and, under favourable conditions, they will collide with each other and coalesce. The rate at which two drops collide is predicted by solving the diffusion equation for Brownian coalescence, and by using a trajectory analysis to follow the relative motion of pairs of drops for gravity-induced coalescence.

The emphasis of our analysis is on the effects of drop interactions on their collision rate, and these are described by the collision efficiency. Since the hydrodynamic resistance to the drop relative motion reduces with a decreasing ratio of the viscosities of the drop fluid and the surrounding fluid, the collision efficiency increases with decreasing viscosity ratio. A qualitative difference in the collision behaviour of viscous drops from that of rigid spheres is demonstrated; finite collision rates for drops are predicted even in the absence of attractive forces, provided that drop deformation is negligible, whereas rigid particles with smooth surfaces will not come into contact in a fluid continuum unless an attractive force is present which is able to overcome the lubrication forces resisting the relative motion. Hydrodynamic interactions between two spherical drops are accounted for exactly by determining the two-sphere relative mobility functions from previous solutions for two drops moving along and normal to their line of centres. These solutions are based on the method of reflections for widely separated drops, lubrication theory for drops in near-contact, and bispherical coordinates for general separations. The hydrodynamic interactions have a greater effect on reducing the rate of gravity collisions than the rate of Brownian collisions.

When a tube, sealed at one end and open to a quiescent environment at the other, is rotated about its axis, fluid flows from the open end along the axis towards the sealed end and returns in an annular boundary layer on the cylindrical wall. This paper describes the first known study to be made of this self-induced flow. Numerical solutions of the Navier–Stokes equations are shown to be in mainly good agreement with experimental results obtained using flow visualization and laser–Doppler anemometry in a rotating glass tube.

The self-induced flow in the tube can be described in terms of the length-to-radius ratio, G, and the Ekman number, E. However, for large values of G (G [ges ] 20), the flow outside the boundary layer on the endwall of the tube can be characterized by a single, modified, Ekman number, E*, where E* = GE. Although most of the fluid entering the open end of the tube is entrained into the annular (Stewartson-type) boundary layer, for small values of E* (E* < 0.2) some flow reaches the sealed end. For this so-called 'short-tube case’, the flow in the boundary layer on the endwall is shown to be similar to that associated with a disk rotating in a quiescent environment: the free disk. The self-induced flow for the short-tube case is believed to be responsible for the ’ hot-poker effect’ used, on some jet engines, to provide ice protection for the nose bullet.

Two-dimensional free-surface flows, which are uniform far upstream in a channel of finite depth that ends suddenly, are computed numerically. The ending is in the form of a vertical wall, which may force the flow upward before it falls down forever as a jet under the effect of gravity. Both subcritical and supercritical solutions are presented. The subcritical solutions are a one-parameter family of solutions, the single parameter being the ratio between the height of the wall and the height of the uniform flow far upstream. On the other hand, the supercritical solutions are a two-parameter family of solutions, the second parameter being the Froude number. Moreover, for some combinations of the parameters, it is shown that the solution is not unique.