In a recent paper Tadjbakhsh & Keller (1960) have predicted that two-dimensional finite standing gravity waves in a rectangular container will have lower frequency than infinitesimal standing waves in deep water but have higher frequency below a certain mean depth to wavelength ratio. This is in strong contrast to the frequency results for finite progressive waves obtained by many investigators. Experimental confirmation of this prediction is reported together with estimates of the magnitude of the frequency effects at several depths. The frequency effect reversal appears to occur at a depth ratio of 0·14, somewhat less than the predicted ratio of 0·17.

The wind-driven circulation of a simple model of the oceanic circulation (linear and homogeneous) is investigated in detail to delineate the role of the Ekman layer mass flux in driving hitherto overlooked components of the oceanic circulation.

The role of upwelling boundary-layer regions in driving interior geostrophic circulations is discussed in detail. Several interesting circulations hidden in the earlier transport theories are described.

Considerable advances have been made in the past few years in treating a variety of problems in slender-body Stokes flow (Taylor 1969; Batchelor 1970; Cox 1970, 1971; Tillett 1970). However, the problem of treating the creeping motion past bluff objects, whose boundaries do not conform to a constant co-ordinate surface of one of the special orthogonal co-ordinate systems for which the Stokes slow-flow equation is simply separable, is still largely unsolved. In the slender-body Stokes flow studies mentioned above, the viscous-flow boundary-value problem is formulated approximately as an integral equation for an unknown distribution of Stokeslets over a line enclosed by the body. The theory is valid for only very extended shapes, since the error in drag decays inversely as the logarithm of the aspect ratio of the object. By contrast, the present authors show that the boundary-value problem for the axisymmetric flow past an arbitrary convex body of revolution can be formulated exactly as an integral equation for an unknown distribution of ring-like singularities over the surface of the body. The kernel in this integral equation is closely related to the fundamental separable solutions of the Stokes slow-flow equation when written in an oblate spheroidal co-ordinate system of vanishing aspect ratio. The two lowest-order appropriate spheroidal singularities are found to provide a complete description for all surface elements, except those perpendicular to the axis. Higher-order singularities of all orders are required to describe axially perpendicular surfaces, such as the ends of a cylinder or the blunt base of an object. The newly derived integral equation is solved numerically to provide the first theoretical solutions for low aspect ratio cylinders and cones. The theoretically predicted drag results are in excellent agreement with experimentally measured values.

A turbulent boundary layer developing on a smooth heated uniform-temperature plate in a zero pressure gradient was set up. The origins of the layers were matched to remove the effect of an In heated starting length. Similarity proposals were tested. The mean flow field followed the usual law of the wall and defect law for both temperature and velocity. Broad-band measurements of stream wise velocity and temperature fluctuations were made, and wall similarity and Townsend's self-preserving flow similarity were found to be applicable, at least after a sufficient flow development.

Some initial attempts to arrive at a comparison between heat and momentum transport are presented. The results include conditionally sampled measurements of instantaneous heat and momentum fluxes and correlations between these two quantities. The fluxes were divided into quadrants. Conditional probabilities and weighted joint probability density functions were measured to determine whether there was a similarity in behaviour of these two fluxes. The concept of ‘hole size’ developed for momentum flux was extended to heat flux and events corresponding to bursts and sweeps in the momentum flux were found to be accompanied by corresponding events in the heat flux.

The dynamics on the laminar–turbulent separatrix is investigated numerically for boundary-layer flows in the subcritical regime. Constant homogeneous suction is applied at the wall, resulting in a parallel asymptotic suction boundary layer (ASBL). When the numerical domain is sufficiently extended in the spanwise direction, the coherent structures found by edge tracking are invariably localized and their dynamics shows bursts that drive a remarkable regular or irregular spanwise dynamics. Depending on the parameters, the asymptotic dynamics on the edge can be either periodic in time or chaotic. A clear mechanism for the regeneration of streaks and streamwise vortices emerges in all cases and is investigated in detail.

In order to understand the flow profiles of complex fluids, a crucial issue concerns the emergence of spatial correlations among plastic rearrangements exhibiting cooperativity flow behaviour at the macroscopic level. In this paper, the rate of plastic events in a Poiseuille flow is experimentally measured on a confined foam in a Hele-Shaw geometry. The correlation with independently measured velocity profiles is quantified by looking at the relationship between the localisation length of the velocity profiles and the localisation length of the spatial distribution of plastic events. To complement the cooperativity mechanisms studied in foam with those of other soft glassy systems, we compare the experiments with simulations of dense emulsions based on the lattice Boltzmann method, which are performed both with and without wall friction. Finally, unprecedented results on the distribution of the orientation of plastic events show that there is a non-trivial correlation with the underlying local shear strain. These features, not previously reported for a confined foam, lend further support to the idea that cooperativity mechanisms, originally invoked for concentrated emulsions (Goyon et al., Nature, vol. 454, 2008, pp. 84–87), have parallels in the behaviour of other soft glassy materials.

The fluid mechanics of heart-valve motion is investigated experimentally and theoretically. From the experiments, it is found that the principal mechanism ensuring optimal mitral-valve closure is the pressure field induced by a strong flow deceleration through the valve. A complete theory, evolved from the Bellhouse & Talbot analytical model, is developed for both the mitral and the aortic valve to provide a relationship between valve motion and valve flow. Results predicted by this theory agree consistently well with those obtained from experiments.

The use of streamwise finlets as a passive flow and aerodynamic noise-control technique is considered in this paper. A comprehensive experimental investigation is undertaken using a long flat plate, and results are presented for the boundary layer and surface pressure measurements for a variety of surface treatments. The pressure–velocity coherence results are also presented to gain a better understanding of the effects of the finlets on the boundary layer structures. The results show that the flow behaviour downstream of the finlets is strongly dependent on the finlet spacing. The use of finlets with coarse spacing leads to a reduction in pressure spectrum at mid- to high frequencies and an increase in spanwise length scale in the trailing-edge region due to flow channelling effects. For the finely distributed finlets, the flow is observed to behave similarly to that of a permeable backward-facing step, with significant suppression of the high-frequency pressure fluctuations but an elevation at low frequencies. Furthermore, the convection velocity is observed to reduce downstream of all finlet treatments. The trailing-edge surface pressure spectrum results have shown that, in order to obtain maximum unsteady pressure reduction, the finlet spacing should be of the order of the thickness of the inner layer of the boundary layer. A thorough study is provided for understanding of the underlying physics of both categories of finlets and their implications for controlling the flow and noise generation mechanism near the trailing edge.

This paper is concerned with deducing the most important features of the steady separated flow past a circular cylinder in the limit of vanishing viscosity. First of all, it is shown that the experimental results reported in an earlier article cannot be reconciled with the notion that, as the Reynolds number Re is increased, the flow becomes inviscid everywhere and that viscous effects remain confined within infinitesimally thin shear layers. In contrast, the limiting solution is visualized as exhibiting three essential features: a viscous, closed ‘wake bubble’ of finite width but with a length increasing linearly with Re in which inertial and viscous effects are everywhere of equal order of magnitude; an outer inviscid flow; and, separating the two regions, a diffuse viscous layer covering large sections of the external field. Further properties of this asymptotic solution include: a finite form drag, a negative rear pressure coefficient at the rear stagnation point of the cylinder, and a Nusselt number for heat transfer which becomes independent of Re along the non-wetted portion of the cylinder surface. This model is shown to be consistent with all the experimental data presently available, including some new heat transfer results that are presented in this paper.

An approximate technique is also proposed which takes into account the asymptotic character of the flow in the vicinity of the cylinder and which predicts the pressure distribution around the cylinder in good agreement with the experiments.

This paper is a direct sequel to Bewley & Aamo (J. Fluid Mech., vol. 499, 2004, pp. 183–196). It was conjectured in that paper, based on the numerical evidence available at that time, that the minimum drag of a constant mass flux channel flow might in fact be that of the laminar flow. This conjecture turned out to be false; Min et al. (J. Fluid Mech., vol. 558, 2006, 309318) discovered a curious control strategy which in fact reduces the time-averaged drag to sub-laminar levels. The present paper establishes rigorously that the power of the control input applied at the walls is always larger than the power saved (due to drag reduction below the laminar level) for any possible control distribution, including that proposed by Min et al. (2006), thus establishing that, energetically (that is accounting for the power saved due to drag reduction and the power exerted by application of the control), the optimal control solution is necessarily to relaminarize the flow.

Using super-large-scale particle image velocimetry (SLPIV), we investigate the spatial structure of the near-wall region in the fully rough atmospheric surface layer with Reynolds number $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{6})$. The field site consists of relatively flat, snow-covered farmland, allowing for the development of a fully rough turbulent boundary layer under near-neutral thermal stability conditions. The imaging field of view extends from 3 m to 19 m above the ground and captures the top of the roughness sublayer and the bottom of an extensive logarithmic region. The SLPIV technique uses natural snowfall as seeding particles for the flow imaging. We demonstrate that SLPIV provides reliable measurements of first- and second-order velocity statistics in the streamwise and wall-normal directions. Our results in the logarithmic region show that the structural features identified in laboratory studies are similarly present in the atmosphere. Using instantaneous vector fields and two-point correlation analysis, we identify vortex structures sharing the signature of hairpin vortex packets. We also evaluate the zonal structure of the boundary layer by tracking uniform momentum zones (UMZs) and the shear interfaces between UMZs in space and time. Statistics of the UMZs and shear interfaces reveal the role of the zonal structure in determining the mean and variance profiles. The velocity difference across the shear interfaces scales with the friction velocity, in agreement with previous studies, and the size of the UMZs scales with wall-normal distance, in agreement with the attached eddy framework.

The lee-wave amplitudes and wave drag for a thin barrier in a two-dimensional stratified flow in which the upstream dynamic pressure and density gradient are constant (Long's model) are determined as functions of barrier height and Froude number for a channel of finite height and for a half-space. Variational approximations to these quantities are obtained and validated by comparison with the earlier results of Drazin & Moore (1967) for the channel and with the results of an exact solution for the half-space, as obtained through separation of variables. An approximate solution of the integral equation for the channel also is obtained through a reduction to a singular integral equation of potential theory. The wave drag tends to increase with decreasing wind speed, but it seems likely that the flow is unstable in the region of high drag. The maximum attainable drag coefficient consistent with stable lee-wave formation appears to be roughly two and almost certainly less than three.

Initially turbulent bluff body wakes decay in the presence of a stable background density gradient to form chains of comparatively stable and long-lived vortex structures, most of the late-time properties of which have been shown to be independent of the initial generating Froude number (for a sphere of diameter, D, moving at speed, U, F = 2U/ND, where N is the buoyancy frequency). Results of experiments with vertical interrogation planes are described, where any anticipated F-dependence might be most evident, as the competing effects of horizontal inertial forcing and the restoring buoyancy force can be measured directly by simultaneous measurement of horizontal and vertical velocity components. Experiments were conducted at sufficiently large values of Re [ges ] 3 × 103 and F [ges ] 4 that turbulence can occur over many scales in the near wake, and the scaling properties might then extrapolate to ocean engineering applications.

When F [ges ] 4, the fluid motions in the intermediate, non-equilibrium régime always occur in coherent patches whose vertical extent is smaller than the total wake height. The patches of vorticity have longer horizontal than vertical coherence lengths, and may be termed layers, even though they are far from uniform in the horizontal. The degree to which the complex vertical structure is later dominated by the mean wake defect depends strongly on F.

The total wake height, LV, depends on the initial value of F so that LV/D ∼ F0.6. LV is established early and remains almost unchanged up to Nt ≈ 30. At later times, the non-equilibrium wake exchanges potential with kinetic energy and re-adjusts according to local dynamical constraints, so that, within each layer, the quasi-two-dimensional flow proceeds without any further dependence on, or memory of, the initial value of F. The flow is everywhere stable to overturning Kelvin–Helmholtz instabilities and local length and velocity scales evolve so that the local horizontal and vertical Froude numbers, FH, FV, are both of order 0.1.

Although Osmidov-length arguments for vertical scale selection appear to be physically appropriate, they do not correctly predict the measured F-dependence in either LV, or in the layer height, lV. Thus the physical mechanism responsible remains elusive, as the alternative laminar instability mechanisms are not presented with the appropriate, scale-free initial conditions over the parameter range in which they have been shown to operate.

Ultimately, the measurements support the application of low FH and FV scaling theories to the late wake flow. The preceding non-equilibrium stage, when the vertical structure of the late wake is determined, does not yield so readily to assumptions involving the smallness of the vertical velocity component.

The shape of a bubble of one liquid inside a denser body of liquid which rotate rigidly together is determined, the effect of gravity being neglected. When the anular velocity of the liquids is zero the bubble assumes a spherical form, and with increasing angular velocity the bubble flattens at the equator and the length increases. It is found that the length of the bubble is asymptotically proportional to the four-thirds power of the angular velocity. If the speed of the rotation is held constant and the volume is increased, then the bubble elongates, the radius approaches a limiting value, and the bubble length increases almost linearly with the volume. This result suggests a method whereby the interfacial surface tension can be measured.

In the second part of the paper the stability of a long bubble subjected to small amplitude axisymmetric disturbances sinusoidal in the axial direction is investigated. The relation between the wave-number and angular velocity for neutral stability is elliptic. When account is taken of the decrease in the radius of the undisturbed bubble with increase in the angular velocity, it is found that the bubble is stable to all wave-lengths provided the radius attains at least 63% of the limiting value. A criterion is then found for the minimum length of the bubble consistent with stability.

The deformation of a Newtonian/viscoelastic drop suspended in a viscoelastic fluid is investigated using a three-dimensional front-tracking finite-difference method. The viscoelasticity is modelled using the Oldroyd-B constitutive equation. Matrix viscoelasticity affects the drop deformation and the inclination angle with the flow direction. Numerical predictions of these quantities are compared with previous experimental measurements using Boger fluids. The elastic and viscous stresses at the interface, polymer orientation, and the elastic and viscous forces in the domain are carefully investigated as they affect the drop response. Significant change in the drop inclination with increasing viscoelasticity is observed; this is explained in terms of the first normal stress difference. A non-monotonic change – a decrease followed by an increase – in the steady-state drop deformation is observed with increasing Deborah number (De) and explained in terms of the competition between increased localized polymer stretching at the drop tips and the decreased viscous stretching due to change in drop orientation angle. The transient drop orientation angle is found to evolve on the polymer relaxation time scale for high De. The breakup of a viscous drop in a viscoelastic matrix is inhibited for small De, and promoted at higher De. Polymeric to total viscosity ratio β was seen to affect the result through the combined parameter βDe indicating a dominant role of the first normal stress difference. A viscoelastic drop in a viscoelastic matrix with matched relaxation time experiences less deformation compared to the case when one of the phases is viscous; but the inclination angle assumes an intermediate value between two extreme cases. Increased drop phase viscoelasticity compared to matrix phase leads to decreased deformation.

In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions.

The existence of an organized streamwise vortical structure, which is superimposed on the well known coherent spanwise vorticity in nominally two-dimensional free shear layers, has been studied extensively. In the presence of stratification, however, buoyancy forces contribute to an additional mechanism for the generation of streamwise vorticity. As the spanwise vorticity layer rolls up and pulls high-density fluid above low-density fluid, a local instability results. The purpose of the current investigation is to force the three-dimensional instability in the stratified shear layer. In this manner, we experimentally observe the effect of buoyancy on the streamwise vortex tube evolution, the evolution of the buoyancy-induced instability, and the interaction between these two vortical structures. A simple numerical model is proposed which captures the relevant physics of the flow evolution. It is found that, depending on the location, streamwise vortices resulting from vortex stretching may be weakened or enhanced by the stratification. Buoyancy-induced vortex structures are shown to form where the unstable part of the interface is tilted by the streamwise vortex tubes. These vortices strengthen initially, then weaken downstream, the timescale for this process depending upon the degree of stratification. For initial Richardson numbers larger than about 0.03, the baroclinically weakened vortex tubes eventually disappear as the flow evolves downstream and the baroclinically generated vortices dominate the three-dimensional flow structure.

The viscously dominated, low-Reynolds-number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behaviour when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low-density fluid through a viscously deformable conduit. These flows are modelled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed-form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviours are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small- to moderate-amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented.

An extension of proper orthogonal decomposition is applied to the wall layer of a turbulent channel flow ($Re_{\unicode[STIX]{x1D70F}}=590$), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the eigenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200–300 viscous wall units. The full spatio-temporal decomposition provides a natural measure of the convection velocity of structures, with a characteristic value of 12$u_{\unicode[STIX]{x1D70F}}$ in the wall layer. Finally, we show that the energy budget can be split into specific contributions for each mode, which provides a closed-form expression for nonlinear effects.

A study of the flow within the critical layer of a forced Rossby-wave is made, using a high-resolution numerical model. The possibility of growth of disturbances through barotropic instability and the extent to which these disturbances modify the subsequent time evolution is of particular interest. The flow is characterized by a parameter μ, equal to the cross-stream lengthscale divided by a downstream wavelength. In the long-wavelength case, μ [Lt ] 1, where there is a clear conceptual division between the instability and the basic flow, the results of the simulation confirm the importance of the growing and saturating disturbances in rearranging the vorticity within the critical layer. When the wavelength is not so long, the distinction between the instability and the straightforward time evolution of the basic flow is less clear. Nonetheless for μ < 0.25 the ultimate evolution is still sensitive to the details of the initial perturbations and in this sense the flow may be regarded as being unstable. The time-integrated absorptivity of the critical layer may be considerably increased by the effects of the instability, sometimes to three or four times that predicted by the Stewartson-Warn-Warn solution. The nature of the flow, at least during the period in which the dynamics are essentially inviscid, does not seem to change when higher harmonics to the forced wave are resonant. The behaviour seen in Béland's (1976) numerical model is re-examined in the light of these findings. A simple model of the redistribution of vorticity by the unstable disturbances is formulated, and its predictions are shown to agree well with the numerical simulations.