A vertically oscillating plate in a boundary layer regulates the vorticity flux rate with respect to time and displaces the vorticity away from the wall. These phenomena are discussed for non-dimensional frequencies of the oscillating plate K = 0, 0.006, 0.01 and 0.02. The velocity data obtained by a split-fibre probe near the wall in the region immediately downstream of the oscillating plate lead to a discussion on the behaviour of the flow structures with respect to the non-dimensional frequency. The physical understanding deduced is complementary to the findings of a smoke-wire flow visualization conducted in this study. An integral analysis of the momentum equation indicates that the mean vorticity flux rate of the present flow is composed of contributions from both the parallel shear layer and the curving streamline. This analysis further suggests that the mean vorticity flux rate can be obtained through a combination of pressure measurements at the wall and in the irrotational region of the flow.

We present a new mechanism of small-scale transition via core dynamics instability (CDI) in an incompressible plane mixing layer, a transition which is not reliant on the presence of longitudinal vortices (‘ribs’) and which can originate much earlier than ribinduced transition. Both linear stability analysis and direct numerical simulation are used to describe CDI growth and subsequent transition in terms of vortex dynamics and vortex line topology. CDI is characterized by amplifying oscillations of core size non-uniformity and meridional flow within spanwise vortices (‘rolls’), produced by a coupling of roll swirl and meridional flow that is manifested by helical twisting and untwisting of roll vortex lines. We find that energetic CDI is excited by subharmonic oblique modes of shear layer instability after roll pairing, when adjacent rolls with out-of-phase undulations merge. Starting from moderate initial disturbance amplitudes, twisting of roll vortex lines generates within the paired roll opposing spanwise flows which even exceed the free-stream velocity. These flows collide to form a nearly irrotational bubble surrounded by a thin vorticity sheath of a large diameter, accompanied by folding and reconnection of roll vortex lines and local transition. We find that accelerated energy transfer to high wavenumbers precedes the development of roll internal intermittency; this transfer, inferred from increased energy at high wavenumbers and an intensification of roll vorticity, occurs prior to the development of strong opposite-signed (to the mean) spanwise vorticity and granularity of the roll vorticity distribution. We demonstrate that these core dynamics are not reliant upon special symmetries and also occur in the presence of moderate-strength ribs, despite entrapment of ribs within pairing rolls. In fact, the roll vorticity dynamics are dominated by CDI if ribs are not sufficiently strong to first initiate transition; thus CDI may govern small-scale transition for moderate initial 3D disturbances, typical of practical situations. Results suggest that CDI constitutes a new generic mechanism for transition to turbulence in shear flows.

In a recent paper Boubnov, Dalziel & Linden (1994) described the response of a stratified fluid to forcing produced by an array of sources and sinks. The sources and sinks were located in a horizontal plane and the flow from the sources was directed horizontally so that fluid was withdrawn from, and re-injected at, its own density level. As a result vertical vorticity was imparted to the fluid with a minimum of vertical mixing. It was found that when the stratification was strong enough to suppress vertical motions an inverse energy cascade was observed leading to the establishment of a large-scale circulation in the fluid. Those experiments were restricted to eight source-sink pairs. The present paper extends this work in two ways. First, up to forty source-sink pairs are used to force the flow, thereby producing a much wider separation of scales between the forcing and the flow domain. An inverse cascade is again found, but in this case the energy transfer to large scales is more rapid. The basic pattern of the large-scale flow is independent of the number of sources but the detailed structure depends on the energy input scale. Second, the effects of rotation about a vertical axis are investigated. It is found that when the Rossby deformation radius exceeds the size of the flow domain, the inverse energy cascade still occurs. However, for smaller values of the deformation scale, which in these experiments are comparable to or smaller than the forcing scale, the inverse cascade is altered by baroclinic instability. When flow structures develop on a scale larger than the deformation scale, usually by the merging of vortices of like sign, these structures are observed to split into smaller vortices of a scale comparable to the deformation scale. The flow appears to evolve with a balance between an anticascade produced by the two-dimensionality of the flow and a cascade due to baroclinic instability. For Rossby radii much smaller than the domain size the flow evolves into finite clumps of vorticity and an asymmetry between anticyclones and cyclones develops. A predominance of coherent anticyclones is observed, and the cyclonic vorticity is contained in more diffuse structures.

Three mathematical models are developed for meniscus roll coating in which there is steady flow of a Newtonian fluid in the narrow gap, or nip, between two contrarotating rolls in the absence of body forces.

The zero flux model predicts a constant pressure gradient within the central core and two eddies, each with an inner structure, in qualitative agreement with observation. The small flux model takes account of a small inlet flux and employs the lubrication approximation to represent fluid velocity as a combination of Couette and Poiseuille flows. Results show that the meniscus coating regime is characterized by small flow rates (λ [Lt ] 1) and a sub-ambient pressure field generated by capillary action at the upstream meniscus. Such flows are found to exist for small modified capillary number, Ca(R/H0)1/2 [lsim ] 0.15, where Ca and R/H0 represent capillary number and the radius to semi-gap ratio, respectively.

A third model incorporates the full effects of curved menisci and nonlinear free surface boundary conditions. The presence of a dynamic contact line, adjacent to the web on the upper roll, requires the imposition of an apparent contact angle and slip length. Numerical solutions for the velocity and pressure fields over the entire domain are obtained using the finite element method. Results are in accord with experimental observations that the flow domain consists of two large eddies and fluid transfer jets or ‘snakes’. Furthermore, the numerical results show that the sub-structure of each large eddy consists of a separatrix with one saddle point, two sub-eddies with centres, and an outer recirculation.

Finally finite element solutions in tandem with lubrication analysis establish the existence of three critical flow rates corresponding to a transformation of the pressure field, the emergence of a ‘secondary snake’ (another fluid transfer jet) and the disappearance of a primary snake.

An experimental study was made of the effect of a periodic velocity perturbation on the separation bubble downstream of the sharp-edged blunt face of a circular cylinder aligned coaxially with the free stream. Velocity fluctuations were produced with an acoustic driver located within the cylinder and a small circumferential gap located immediately downstream of the fixed separation line to allow communication with the external flow. The flow could be considerably modified when forced at frequencies lower than the initial Kelvin-Helmholtz frequencies of the free shear layer, and with associated vortex wavelengths comparable to the bubble height. Reattachment length, bubble height, pressure at separation, and average pressure on the face were all reduced. The effects on the large-scale structures were studied using flow photographs obtained by the smoke-wire technique. The forcing increased the entrainment near the leading edge. It was concluded that the final vortex of the shear layer before reattachment is an important element of the flow structure. There are two different instabilities involved: the Kelvin-Helmholtz instability of the free shear layer and the ‘shedding’-type instability of the entire bubble. The former consists of an interaction of the shear layer vorticity with itself, the latter with its images that result because of the presence of a wall. In order to determine the optimum forcing frequency, a method of frequency scaling is proposed which correlates data for a variety of bubbles and supports an analogy with Kármán vortex shedding.

A novel method is introduced for solving the three-dimensional Stokes equations via a spectral element approach to the boundary integral method. The accuracy and convergence of the method are illustrated through applications involving rigid particles, deformable droplets and interacting particles. New physical results are obtained for two applications in low Reynolds number flow: the permeability of periodic models of a porous membrane and the instability of a toroidal droplet subject to non-axisymmetric perturbations. Further applications are described in the companion paper (Higdon & Muldowney 1995).

Numerical computations are performed to evaluate the resistance functions for low Reynolds number flow past spherical particles, droplets and bubbles in cylindrical domains. Spheres of arbitrary radius a and radial position b move with arbitrary velocity U within a cylinder of radius R. The undisturbed fluid may be at rest, or subject to a pressure-driven flow with maximum velocity U0. The spectral boundary element method is employed to compute the resistance force for torque-free bodies in three cases: rigid solids, fluid droplets with viscosity ratio λ = 1, and bubbles with viscosity ratio λ = 0. A lubrication theory is developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions are developed which accurately represent the numerical data over the entire range of particle positions 0 < b/(R − a) < 1 for all particle sizes in the range 0 < a/R < 0.9. The resistance functions are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels.

The three-dimensional development of controlled transition in a flat-plate boundary layer is investigated by direct numerical simulation (DNS) using the complete Navier-Stokes equations. The numerical investigations are based on the so-called spatial model, thus allowing realistic simulations of spatially developing transition phenomena as observed in laboratory experiments. For solving the Navier-Stokes equations, an efficient and accurate numerical method was developed employing fourth-order finite differences in the downstream and wall-normal directions and treating the spanwise direction pseudo-spectrally. The present paper focuses on direct simulations of the wind-tunnel experiments by Kachanov et al. (1984, 1985) of fundamental breakdown in controlled transition. The numerical results agreed very well with the experimental measurements up to the second spike stage, in spite of relatively coarse spanwise resolution. Detailed analysis of the numerical data allowed identification of the essential breakdown mechanisms. In particular, from our numerical data, we could identify the dominant shear layers and vortical structures that are associated with this breakdown process.

When the resonance condition is satisfied, i.e. that the local group velocity of short surface waves matches the local velocity associated with a larger-scale surface flow, it is known that the short waves are reflected or trapped by the flow. A typical example is the case of short surface waves propagating on long surface waves. By direct numerical resolution of the kinematic equations, some aspects of the reflection or trapping are first examined. Next, the effects of a second long wave on the trajectory of the short waves are considered. It is found that the trajectory is strongly distorted in general. Reflection still occurs, having a larger effect on the variation of the shortwave wavenumber than when only a single long wave is present. The entrapment becomes more sporadic. At short time intervals, a forced Mathieu equation is found to govern the short-wave development. This leads to a discussion on a more general physical context.

Peristaltic motion of two immiscible viscous incompressible fluids in a circular tube is studied in pumping and copumping ranges under long-wavelength and low-Reynoldsnumber assumptions. The effect of the peripheral-layer viscosity on the time-averaged flux and the mechanical efficiency is studied. The formation and growth of the trapping zone in the core and the peripheral layer are explained. It is observed that the bolus volume in the peripheral layer increases with an increase in the viscosity ratio. The limits of the time-averaged flux $\bar{Q}$ for trapping in the core are obtained. The trapping observed in the peripheral layer decreases in size with an increase in $\bar{Q}$ but never disappears. The development of the complete trapping of the core fluid by the peripheral-layer fluid with an increase in the time-averaged flux is demonstrated. The effect of peripheral-layer viscosity on the reflux layer is investigated. It is also observed that the reflux occurs in the entire pumping range for all viscosity ratios and it is absent in the entire range of copumping.

The destabilization of a thin three-dimensional non-wetting film above a solid wall is examined for the special case in which surfactant is adsorbed onto the free surface of the film. Attention is restricted to the case of a Newtonian surface, with surfactant displaying rapid surface diffusion or exhibiting small Marangoni number, such that the dominant intrinsic interfacial stress is of a purely viscous origin. A surface-excess force approach is adopted for the purpose of incorporating into the analysis the attractive/repulsive dispersive forces acting between the solid wall and the film. Three coupled nonlinear partial differential equations are obtained that describe the ‘large-wavelength’ spatio-temporal evolution of the free film surface following a small initial disturbance. These equations are shown to reduce to results in the literature in the limit of zero interfacial viscosities. Employing linear stability analysis, an explicit dispersion equation is obtained relating the growth coefficient to interfacial viscosities. It is found, at least in the linear regime, that the sum of interfacial shear and dilatational viscosities – and not each separately – imparts a damping effect that in the most extreme case is four-fold relative to the case of no interfacial viscosities. Nonlinear stability analysis in the limiting case of a two-dimensional film indicates that interfacial viscosities may strongly hinder the onset of instability through large interfacial stresses that arise in the vicinity of trough and crest regions of the film.

The Ekman boundary layer scalings associated with small oscillatory flows in a rapidly rotating system are well known to break down at the critical latitudes. The effect of these anomalous boundary regions on the interior flow has long been an issue of some concern. We argue, using analytical results derived in a model problem based on Stewartson's (1957) split-disk configuration, that these boundary regions spawn internal shear layers. These shear layers have their natural length scale of E1/3 and carry velocities only O(E1/6) smaller than the Rossby number – where E is the Ekman number – if the boundary is concave away from the fluid at the critical latitude. Otherwise, only a weaker shear layer is produced of length scale E1/5 containing velocities of O(E3/10) smaller than the Rossby number rather than the usual O(E1/5) value for viscous flows in the interior. We show how these layers can carry angular momentum by themselves or more significantly in combination with the interior inviscid flow, so as to influence the mean flow at leading order. These conclusions are then discussed in the context of a precessing oblate spheroid of fluid for which detailed experimental observations are available.

We present a direct numerical solution of a particular inertial oscillation, the so-called 'spin-over’ mode, in spherical geometry. This mode is particularly relevant to the fluid flow within a precessing oblate spheroid. We demonstrate that the oscillatory Ekman layer breaks down at ±30° latitude, and that this breakdown spawns internal shear layers. We show that the structure of these shear layers is different for a full sphere and a spherical shell, as noted in the preceding paper (Kerswell 1995). Despite the existence of these shear layers, however, the numerical decay rates agree to within 1 % with the asymptotic decay rates, which neglect any possible shear layers. Finally, we consider the nonlinear mean flow profiles driven by this mode, and demonstrate that our numerical results agree reasonably well with experimental results.

We investigate the oscillation spectrum of electromagnetically levitated metal droplets. In the case of electromagnetic levitation, gravity is compensated by a Lorentz force, which is generated by an external current. The oscillation spectrum contains information about the thermophysical properties of the liquid metal, namely surface tension and viscosity. For a correct interpretation of these spectra the influence of the external forces on the frequencies and the damping of the surface waves must be well understood. The external forces deform the droplet, so that the static equilibrium shape is aspherical. For a perfect conductor the effect of the Lorentz force and gravity on the oscillation spectrum is calculated for an arbitrary magnetic field and arbitrary values of the viscosity. The high Reynolds number limit is evaluated. Explicit results are obtained for a linear magnetic field, which describes the experimental situation well.

The attached eddy hypothesis developed for zero pressure gradient boundary layers and for pipe flow is extended here to boundary layers with arbitrary streamwise pressure gradients, both favourable and adverse. It is found that in order to obtain the correct quantitative results for all components of the Reynolds stresses, two basic types of eddy structure geometries are required. The first type, called type-A, is interpreted to give a ‘wall structure’ and the second, referred to as type-B, gives a ‘wake structure’. This is in analogy with the conventional mean velocity formulation of Coles where the velocity is decomposed into a law of the wall and a law of the wake.

If the above mean velocity formulation is accepted, then in principle, once the eddy geometries are fixed for the two eddy types, all Reynolds stresses and associated spectra contributed from the attached eddies can be computed without any further empirical constants. This is done by using the momentum equation and certain convolution integrals developed here based on the attached eddy hypothesis. The theory is developed using data from equilibrium and quasi-equilibrium flows. In Part 2 the authors’ non-equilibrium data are used.

In Part 1 an extension of the attached eddy hypothesis was developed and applied to equilibrium pressure gradient turbulent boundary layers. In this paper the formulation is applied to data measured by the authors from non-equilibrium layers and agreement with the extended theory is encouraging. Also power spectra of the Reynolds stresses as developed from the extended theory compare favourably with experiment. The experimental data include a check of cone-angle effects by using a flying hot wire.