An experimental and theoretical investigation of turbulent boundary layers developing in a sink-flow pressure gradient was undertaken. Three flow cases were studied, corresponding to different acceleration strengths. Mean-flow measurements were taken for all three cases, while Reynolds stresses and spectra measurements were made for two of the flow cases. In this study attention was focused on the evolution of the layers to an equilibrium turbulent state. All the layers were found to attain a state very close to precise equilibrium. This gave equilibrium sink flow data at higher Reynolds numbers than in previous experiments. The mean velocity profiles were found to collapse onto the conventional logarithmic law of the wall. However, for profiles measured with the Pitot tube, a slight ‘kick-up’ from the logarithmic law was observed near the buffer region, whereas the mean velocity profiles measured with a normal hot wire did not exhibit this deviation from the logarithmic law. As the layers approached equilibrium, the mean velocity profiles were found to approach the pure wall profile and for the highest level of acceleration Π was very close to zero, where Π is the Coles wake factor. This supports the proposition of Coles (1957), that the equilibrium sink flow corresponds to pure wall flow. Particular interest was also given to the evolutionary stages of the boundary layers, in order to test and further develop the closure hypothesis of Perry, Marusic & Li (1994). Improved quantitative agreement with the experimental results was found after slight modification of their original closure equation.

A scenario of transition to turbulence likely to occur during the development of natural disturbances in a flat-plate boundary layer is studied. The perturbations at the leading edge of the flat plate that show the highest potential for transient energy amplification consist of streamwise aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. Direct numerical simulations are used to follow the nonlinear evolution of these streaks and to verify secondary instability calculations. The theory is based on a linear Floquet expansion and focuses on the temporal, inviscid instability of these flow structures. The procedure requires integration in the complex plane, in the coordinate direction normal to the wall, to properly identify neutral modes belonging to the discrete spectrum. The streak critical amplitude, beyond which streamwise travelling waves are excited, is about 26% of the free-stream velocity. The sinuous instability mode (either the fundamental or the subharmonic, depending on the streak amplitude) represents the most dangerous disturbance. Varicose waves are more stable, and are characterized by a critical amplitude of about 37%. Stability calculations of streamwise streaks employing the shape assumption, carried out in a parallel investigation, are compared to the results obtained here using the nonlinearly modified mean fields; the need to consider a base flow which includes mean flow modification and harmonics of the fundamental streak is clearly demonstrated.

Rotating Boussinesq convection in a plane layer is governed by two dimensionless groups in addition to the Rayleigh number R: the Prandtl number σ and the Taylor number Ta. Scaled equations for fully nonlinear rotating convection in the limit of rapid rotation and small Prandtl number, where the onset of convection is oscillatory, are derived by considering distinguished limits where σnTa1/2 ∼ 1 but σ → 0 and Ta → ∞, for different n > 1. In the resulting asymptotic expansion in powers of Ta−1/2 and the amplitude of convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of the subdominant buoyancy and inertial terms. For the most interesting case, n = 4, the stability of different planforms near onset is investigated using a double expansion in powers of Ta−1/8 and the amplitude of convection ε. The lack of a buoyancy term at leading order demands that the perturbation expansion be continued through six orders to derive amplitude equations determining the dynamics. The case n = 1 is also analysed. The relevance of this theory to experimental results is briefly discussed.

Steady, finite-amplitude internal-wave disturbances, induced by nearly hydrostatic stratified flow over locally confined topography that is more elongated in the spanwise than the streamwise direction, are discussed. The nonlinear three-dimensional equations of motion are handled via a matched-asymptotics procedure: in an ‘inner’ region close to the topography, the flow is nonlinear but weakly three-dimensional, while far upstream and downstream the ‘outer’ flow is governed, to leading order, by the fully three-dimensional linear hydrostatic equations, subject to matching conditions from the inner flow. Based on this approach, non-resonant flow of general (stable) stratification over finite-amplitude topography in a channel of finite depth is analysed first. Three-dimensional effects are found to inhibit wave breaking in the nonlinear flow over the topography, and the downstream disturbance comprises multiple small-amplitude oblique wavetrains, forming supercritical wakes, akin to the supercritical free-surface wake induced by linear hydrostatic flow of a homogeneous fluid. Downstream wakes of a similar nature are also present when the flow is uniformly stratified and resonant (i.e. the flow speed is close to the long-wave speed of one of the modes in the channel), but, in this instance, they are induced by nonlinear interactions precipitated by three-dimensional effects in the inner flow and are significantly stronger than their linear counterparts. Finally, owing to this nonlinear-interaction mechanism, vertically unbounded uniformly stratified hydrostatic flow over finite-amplitude topography also features downstream wakes, in contrast to the corresponding linear disturbance that is entirely locally confined.

The unsteady evolution of a boundary layer induced by a rectilinear vortex convecting above a heated surface is considered numerically. This model problem is representative of the types of interactions that can occur when vortices encounter solid surfaces in a wide variety of diverse applications involving high-Reynolds-number and high-Grashof-number flows. It is known that in the case without heat transfer, the vortex-induced boundary layer evolves toward a singularity as it forms a sharp spike that erupts away from the surface. Numerical solutions of the unsteady mixed-convection boundary-layer equations in the Boussinesq limit are obtained in Lagrangian coordinates. Solutions for various values of the inclination angle of the surface and Grashof number show that the coupling between the fluid flow and heat transfer can have a dramatic effect upon the transport of momentum and energy within the boundary layer induced by the vortex. The unsteady eruption convects high-temperature, near-wall fluid away from the surface and causes large gradients in the thermal boundary layer. The buoyancy force acting on the heated boundary-layer fluid can also have a significant impact on the unsteady separation process, either accelerating or delaying it, depending upon the inclination angle of the surface.

The flow patterns in the steady, viscous flow in a cylinder with a rotating bottom and a free surface are investigated by a combination of topological and numerical methods. Assuming the flow is axisymmetric, we derive a list of possible bifurcations of streamline structures on varying two parameters, the Reynolds number and the aspect ratio of the cylinder. Using this theory we systematically perform numerical simulations to obtain the bifurcation diagram. The stability limit for steady flow is found and established as a Hopf bifurcation. We compare with the experiments by Spohn, Mory & Hopfinger (1993) and find both similarities and differences.

We present results from a direct numerical simulation of the passive transport of solid particles by a fully developed turbulent channel flow with a Reynolds number of 180 based on the friction velocity and the channel half-width. Three particle sets are studied, ranging in diameter from 0.5 to 1.4 viscous wall units and in aerodynamic time constant from 0.6 to 56 centreline Kolmogorov time scales. We use particle number density histograms and fractal dimensions to show that the level of order in the particle spatial distribution peaks near a Stokes number of unity based on the Kolmogorov time scale. We then quantify the relationship between this spatial distribution and the instantaneous flow topology. The results indicate that the previously reported preferential concentration of particles in low-speed streaks leads to a suppression of particle velocities in the viscous sublayer and buffer region even in the presence of streamwise gravitational acceleration. In other regions of the flow, the particles' non-random spatial distribution is shown to be uncorrelated with the local flow topology. We compare our results with the experimental data of Kulick et al. (1994) and Fessler et al. (1994) and confirm that the latter authors' results were not influenced by turbulence modification.

The wetting kinetics of a drop on a solid surface is measured by observing the movement of the contact line, which is often seen to be unstable, showing a scalloped profile. Many factors have been cited, which, although they can cause instability, can also be eliminated from the experiments, but still the instabilities appear. The basic shape of a spreading drop has a large curvature localized in the vicinity of the contact line as determined by microscopy. It is shown here using linear stability analysis that this curvature can destabilize the contact line region. When the drop profile is disturbed from a basic thickness of h to h + h′, there are two contributions from h′ in the form of added Laplace pressure. One of these is commonly accounted for in the stability analyses. The other is not, and occurs only if the basic shape has a curvature, and the drop has a large curvature near the apparent dynamic contact line, but only for a wetting liquid. This is why instability is not reported in the case of spreading of drops of non-wetting liquids. It also explains why instability gives rise to the changed spreading kinetics of drops that are sometimes reported in the literature, and suggests that as larger curvatures are expected in forced spreading those cases are probably accompanied quite frequently by unstable contact lines.

Bypass transition in an initially laminar boundary layer beneath free-stream turbulence is simulated numerically. New perspectives on this phenomenon are obtained from the numerical flow fields. Transition precursors consist of long backward jets contained in the fluctuating u-velocity field; they flow backwards relative to the local mean velocity. The jets extend into the upper portion of the boundary layer, where they interact with free-stream eddies. In some locations a free-stream perturbation to the jet shear layer develops into a patch of irregular motion – a sort of turbulent spot. The spot spreads longitudinally and laterally, and ultimately merges into the downstream turbulent boundary layer. Merging spots maintain the upstream edge of the turbulent region. The jets, themselves, are produced by low-frequency components of the free-stream turbulence that penetrate into the laminar boundary layer. Backward jets are a component of laminar region streaks.

A method to construct turbulent inflow from Orr–Sommerfeld continuous modes is described. The free-stream turbulent intensity was chosen to correspond with the experiment by Roach & Brierly (1990). Ensemble-averaged numerical data are shown to be in good agreement with laboratory measurements.

The dynamics of the evolution of turbulence statistics depend on the structure of the turbulence. For example, wavenumber anisotropy in homogeneous turbulence is known to affect both the interaction between large and small scales (Kida & Hunt 1989), and the non-local effects of the pressure–strain-rate correlation in the one-point Reynolds stress equations (Reynolds 1989; Cambon et al. 1992). Good quantitative measures of turbulence structure are easy to construct using two-point or spectral data, but one-point measures are needed for the Reynolds-averaged modelling of engineering flows. Here we introduce a systematic framework for exploring the role of turbulence structure in the evolution of one-point turbulence statistics. Five one-point statistical measures of the energy-containing turbulence structure are introduced and used with direct numerical simulations to analyse the role of turbulence structure in several cases of homogeneous and inhomogeneous turbulence undergoing diverse modes of mean deformation. The one-point structure tensors are found to be useful descriptors of turbulence structure, and lead to a deeper understanding of some rather surprising observations from DNS and experiments.

The bubbly two-phase flow and electric current density distribution along a single, vertical, gas-evolving electrode are modelled and the results of a boundary layer analysis are presented. Existing empirical models for particle transport in sheared and sedimenting suspensions are adopted for the bubble mixture to close the two-phase model. Ionic species concentrations are shown to be essentially homogeneous as the mixing effect of the bubble suspension usually is much larger than dispersion by molecular diffusion even at laminar flow conditions. The non-uniformity of the bubble distribution along the electrode results in a non-uniform current density distribution, which agrees well with existing experimental findings in the literature.

This paper describes an experimental study of two-dimensional surface water waves propagating on a depth-varying current with a non-uniform vorticity distribution. The investigation is divided into two parts. The first concerns the ‘equilibrium’ conditions in which the oscillatory wave motion and the current co-exist. Measurements of the water-surface elevation, the water-particle kinematics, and the near-bed pressure fluctuations are compared to a number of wave and wave–current solutions including a nonlinear model capable of incorporating the vertical structure of the current profile. These comparisons confirm that the near-surface vorticity leads to an important modification of the dispersion equation, and thus affects the nature of the wave-induced orbital motion over the entire water depth. However, the inclusion of vorticity-dependent terms within the dispersion equation is not sufficient to define the combined wave–current flow. The present results suggest that vorticity may lead to a significant change in the water-surface profile. If a current is positively sheared, dU/dz > 0, with negative vorticity at the water surface, as would be the case in a wind-driven current, a wave propagating in the same direction as the current will experience increased crest–trough asymmetry due to the vorticity distribution. With higher and sharper wave crests there is a corresponding increase in both the maximum water-particle accelerations and the maximum horizontal water-particle velocities. These results are consistent with previous theoretical calculations involving uniform vorticity distributions (Simmen & Saffman 1985 and Teles da Silva & Peregrine 1988).

The second part of the study addresses the ‘gradually varying’ problem in which there are changes in the current, the wavelength and the wave height due to the initial interaction between the wave and the current. These data show that there is a large and non-uniform change in the current profile that is dependent upon both the steepness of the waves and the vorticity distribution. Furthermore, comparisons between the measured wave height change and a number of solutions based on the conservation of wave action, confirm that the vorticity distribution plays a dominant role. In the absence of a conservation equation for wave action appropriate for nonlinear waves on a depth-varying current, an alternative approach based on the conservation of total energy flux, first proposed by Longuet-Higgins & Stewart (1960), is shown to be in good agreement with the measured data.

The unsteady behaviour of an infinitely long fluid-loaded elastic plate subject to a single-frequency line forcing in the presence of a uniform mean flow is known to exhibit a number of interesting phenomena. These include the onset of absolute instability for non-dimensional flow speeds U in excess of some critical speed Uc, and various interesting propagation effects when U < Uc. In the latter respect Crighton & Oswell (1991) have shown that over a particular frequency range there exists an anomalous neutral mode with group velocity directed towards the driver, in violation of the usual Lighthill outgoing radiation condition. Similar results have been found by Peake (1997) when transverse curvature effects are included. In this paper we seek to extend these results and consider the substantially harder problem of a fluid-loaded elastic plate with uniform mean flow which is subject to a point forcing, thereby resulting in a two-dimensional structural problem. A systematic method for determining the absolute instability threshold is developed, and it is shown that the flow is absolutely unstable for flow speeds U > Uc, where Uc is the one-dimensional value found by Crighton & Oswell. At flow speeds U < Uc the flow is marginally stable and convective growth is found to occur downstream of the driver, over a particular frequency range depending on the transverse Fourier wavenumber ky, within a wedge-shaped region. Outside this wedge-shaped region there is only neutral mode behaviour. Asymptotic forms are found for the dominant large-distance causal flexion response downstream of the driver inside and outside the wedge region, and the appropriate critical angle for the wedge region is identified. Within the convective instability wedge the flexion and critical angle take two different forms depending on whether the frequency ω is greater or less than U2/√5. In addition to this interesting behaviour, the flow also exhibits the usual anomalous neutral mode behaviour and, as with Peake's problem, we also find an extra stability (hoop) resulting in neutral mode behaviour over a small frequency range. Asymptotic forms are also found for the threshold frequencies which divide up the various regions of stability of the system (neutral, neutral anomalous, convectively unstable), as a function of ky, and are compared with the results of both Crighton & Oswell and Peake.

The steady response of a fluid consisting of two regions of different density, the lower of which is of finite depth, is considered during withdrawal. Super-critical flows are considered in which water from both layers is being withdrawn, meaning that the interface is drawn down directly into the sink. The results indicate that if the flow rate is above some minimum, the angle of entry of the interface depends more strongly on the relative depth of the sink than on the flow rate. This has quite dramatic consequences for the understanding of selective withdrawal from layered fluids.

The results of a laboratory experiment designed to study turbulent entrainment at sheared density interfaces are described. A stratified shear layer, across which a velocity difference ΔU and buoyancy difference Δb is imposed, separates a lighter upper turbulent layer of depth D from a quiescent, deep lower layer which is either homogeneous (two-layer case) or linearly stratified with a buoyancy frequency N (linearly stratified case). In the parameter ranges investigated the flow is mainly determined by two parameters: the bulk Richardson number RiB = ΔbD/ΔU2 and the frequency ratio fN = ND=ΔU.

When RiB > 1.5, there is a growing significance of buoyancy effects upon the entrainment process; it is observed that interfacial instabilities locally mix heavy and light fluid layers, and thus facilitate the less energetic mixed-layer turbulent eddies in scouring the interface and lifting partially mixed fluid. The nature of the instability is dependent on RiB, or a related parameter, the local gradient Richardson number Rig = N2L/ (∂u/∂z)2, where NL is the local buoyancy frequency, u is the local streamwise velocity and z is the vertical coordinate. The transition from the Kelvin–Helmholtz (K-H) instability dominated regime to a second shear instability, namely growing Hölmböe waves, occurs through a transitional regime 3.2 < RiB < 5.8. The K-H activity completely subsided beyond RiB ∼ 5 or Rig ∼ 1. The transition period 3.2 < RiB < 5 was characterized by the presence of both K-H billows and wave-like features, interacting with each other while breaking and causing intense mixing. The flux Richardson number Rif or the mixing efficiency peaked during this transition period, with a maximum of Rif ∼ 0.4 at RiB ∼ 5 or Rig ∼ 1. The interface at 5 < RiB < 5.8 was dominated by ‘asymmetric’ interfacial waves, which gradually transitioned to (symmetric) Hölmböe waves at RiB > 5:8.

Laser-induced fluorescence measurements of both the interfacial buoyancy flux and the entrainment rate showed a large disparity (as large as 50%) between the two-layer and the linearly stratified cases in the range 1.5 < RiB < 5. In particular, the buoyancy flux (and the entrainment rate) was higher when internal waves were not permitted to propagate into the deep layer, in which case more energy was available for interfacial mixing. When the lower layer was linearly stratified, the internal waves appeared to be excited by an ‘interfacial swelling’ phenomenon, characterized by the recurrence of groups or packets of K-H billows, their degeneration into turbulence and subsequent mixing, interfacial thickening and scouring of the thickened interface by turbulent eddies.

Estimation of the turbulent kinetic energy (TKE) budget in the interfacial zone for the two-layer case based on the parameter α, where α = (−B + ε)/P, indicated an approximate balance (α ∼ 1) between the shear production P, buoyancy flux B and the dissipation rate ε, except in the range RiB < 5 where K-H driven mixing was active.

Eady's model of baroclinic instability has been generalized by including β (the meridional gradient of planetary potential vorticity) while assuming that total potential vorticity is uniform. Moreover, the problems of Eady and of Phillips have been enriched by including a fixed topography or a free boundary (which implies a flow-dependent geostrophic topography). The most general cases (with β, fixed topography and a free boundary) of both problems are shown to have nearly identical stability properties, mainly determined by two Charney numbers: the planetary one and a topographic one. The question of whether this generalized baroclinic instability problem can be described by wave resonance or component ‘resonance’ is addressed. By waves are meant physical modes, which could freely propagate by themselves but are effectively coupled by an independent basic shear, producing the instability. Components, on the other hand, are mathematical modes for which the shear is also crucial for their existence, not just for their coupling, hence the quotation marks around ‘resonance’. In this paper it is shown that both scenarios, components ‘resonance’ and waves resonance, cast light on the free-boundary baroclinic instability problem by providing explanations of the instability onset (at minimum shear) and maximum growth rate cases, respectively. The importance of the mode pseudomomentum for the fulfillment of both mechanisms is also stressed.