We consider the stability of a rectilinear liquid region whose boundary is composed of a solid cylindrical substrate of arbitrary shape and a free surface whose cross-section, in the absence of gravity, is a circular arc. The liquid–solid contact angle is a prescribed material property. A variational technique, using an energy functional, is developed that predicts the minimum wavelength for transverse instability under the action of capillarity. Conversely, certain configurations are absolutely stable and a simple stability criterion is derived. Stability is guaranteed if, for given substrate geometry and given contact angle, the unperturbed meniscus pressure is an increasing function of the liquid cross-sectional area. The analysis is applied to a variety of liquid/substrate configurations including (i) a liquid ridge with contact lines pinned to the sharp edges of a slot or groove, (ii) liquid ridges with free contact lines on flat and wedge-shaped substrates as well as substrates of circular or elliptical cross-section. Results are consistent with special cases previously treated including those that employ a slope-small-slope approximation.

Carefully timed tandem microbubbles have been shown to produce directional and targeted membrane poration of individual cells in microfluidic systems, which could be of use in ultrasound-mediated drug and gene delivery. This study aims at contributing to the understanding of the mechanisms at play in such an interaction. The dynamics of single and tandem microbubbles between two parallel plates is studied numerically and analytically. Comparisons are then made between the numerical results and the available experimental results. Numerically, assuming a potential flow, a three-dimensional boundary element method (BEM) is used to describe complex bubble deformations, jet formation, and bubble splitting. Analytically, compressibility and viscous boundary layer effects along the channel walls, neglected in the BEM model, are considered while shape of the bubble is not considered. Comparisons show that energy losses modify the bubble dynamics when the two approaches use identical initial conditions. The initial conditions in the boundary element method can be adjusted to recover the bubble period and maximum bubble volume when in an infinite medium. Using the same conditions enables the method to recover the full dynamics of single and tandem bubbles, including large deformations and fast re-entering jet formation. This method can be used as a design tool for future tandem-bubble sonoporation experiments.

Evolution of Lagrangian coherent structures (LCS) in a flat plate boundary layer transition induced by the wake of a circular cylinder is investigated. Both hydrogen bubble visualization and particle image velocimetry (PIV) techniques are used. It is found that downstream of the cylinder, the disturbance in the boundary layer experiences a fast growth followed by a slow decay in the transition. Lagrangian coherent structures are revealed by qualitative hydrogen bubble visualizations and quantitative finite-time Lyapunov exponents (FTLE) fields derived from the PIV data. The evolution of the LCS is considered from the very beginning of the transition up to when the boundary layer becomes fully developed turbulent flow. The mean convection velocity and average inclination angle of the LCS are first extracted from the FTLE fields. The streamwise length of the low-speed streaks seems to increase, while their spanwise distance decreases in the boundary layer transition. Proper orthogonal decomposition (POD) of the PIV data shows that low-speed streaks associated with the hairpin vortices and hairpin packets are the dominant coherent structures close to the wall in the transitional and turbulent boundary layer. The POD modes also reveal a variety of scales in the turbulent boundary layer. Moreover, it is found that large-scale coherent structures can modulate the amplitude of the small-scale ones.

Processing the data from a large variety of zero-pressure-gradient boundary layer flows shows that the Reynolds-number-dependent scaling law, which the present authors obtained earlier for pipes, gives an accurate description of the velocity distribution in a self-similar intermediate region adjacent to the viscous sublayer next to the wall. The appropriate length scale that enters the definition of the boundary layer Reynolds number is found for all the flows under investigation.

Another intermediate self-similar region between the free stream and the first intermediate region is found under conditions of weak free-stream turbulence. The effects of turbulence in the free stream and of wall roughness are assessed, and conclusions are drawn.

The 50th Euromech Colloquium, on wind-tunnel simulation of the atmospheric boundary layer, was held in Berlin from 23–25 September 1974. Thirty-eight participants from eleven countries were present. Papers were presented describing and analysing different methods of simulation of neutral, stable and unstable atmospheric conditions in various types of wind tunnel. Numerous applications of wind-tunnel simulations were described or mentioned in the papers and the discussion sessions. Some conclusions about the validity, the techniques, the limitations and future developments of wind-tunnel simulations were reached in discussion. Tables are presented in appendix A listing the institutes in Europe and the U.S.A. of actual or invited participants where wind tunnels are used for simulation work; also listed are the characteristics of the wind tunnels and relevant measurements of the simulated atmospheric boundary layers, to enable comparisons to be made between different techniques.

We show that the rippling often observed on small progressive gravity waves can be explained in terms of a nearly resonant harmonic nonlinear interaction. The resonance condition is that the phase speeds of the two waves must be nearly identical. The in viscid analysis is generalized to any order in a small parameter proportional to the wave steepness. Wave tank measurements provide experimental evidence for most of the predicted results. The phenomenon of resonant rippling is further shown to be not just peculiar to capillary-gravity waves, but in fact possible for any weakly nonlinear dispersive wave system whose dispersion relation has discrete pairs of solutions nearly satisfying the resonance conditions.

A linear stability analysis and experiments were carried out for the laminar–turbulent transition of three-dimensional incompressible boundary layers induced on the surface of a cone rotating around the axis of symmetry with constant angular speed in still fluid. Five cones having total angle of 30°–150° were tested. The results show that the critical and transition Reynolds numbers, the direction of spiral vortices appearing in the transition region and their number on a cone increase as the cone angle is increased, and they tend to the values for the case of a rotating disk. Flow visualizations were made for the transitional process and also for cross-sectional flows of spiral vortices.

Thermal convection of a Boussinesq fluid in a layer confined between two infinite horizontal walls is investigated by direct numerical simulation (DNS) and by large-eddy simulation (LES) for zero horizontal mean motion. The lower-surface height varies sinusoidally in one horizontal direction while remaining constant in the other. Several cases are considered with amplitude δ up to 0.15H and wavelength λ of H to 8H (inclination up to 43°), where H is the mean fluid-layer height. Constant heat flux is prescribed at the lower surface of the initially at rest and isothermal fluid layer. In the LES, the surface is treated as rough surface (z0 = 10−4H) using the Monin-Oboukhov relationships. At the flat top an adiabatic frictionless boundary condition is applied which approximates a strong capping inversion of an atmospheric convective boundary layer. In both horizontal directions, the model domain extends over the same length (either 4H or 8H) with periodic lateral boundary conditions.

We compare DNS of moderate turbulence (Reynolds number based on H and on the convective velocity is 100, Prandtl number is 0.7) with LES of the fully developed turbulent state in terms of turbulence statistics and Characteristic large-scale-motion structures. The LES results for a flat surface generally agree well with the measurements of Adrian et al. (1986). The gross features of the flow statistics, such as profiles of turbulence variances and fluxes, are found to be not very sensitive to the variations of wavelength, amplitude, domain size and resolution and even the model type (DNS or LES), whereas details of the flow structure are changed considerably. The LES shows more turbulent structures and larger horizontal scales than the DNS. To a weak degree, the orography enforces rolls with axes both perpendicular and parallel to the wave crests and with horizontal wavelengths of about 2H to 4H. The orography has the largest effect for λ = 4H in the LES and for λ = 2H in the DNS. The results change little when the size of the computational domain is doubled in both horizontal directions. Most of the motion energy is contained in the large-scale structures and these structures are persistent in time over periods of several convective time units. The motion structure persists considerably longer over wavy terrain than over flat surfaces.

The flow structure associated with a line bubble plume in shallow water is investigated. G. I. Taylor has proposed the use of such plumes as wavebreakers. To be effective the surface current generated should be stable for a distance of the order of the wavelength, which in turn could be several times the depth. It appears that the formation of recirculating cells can affect the wavebreaking potential of line bubble plumes. The paper presents observations and measurements of the cell structure associated with line bubble plumes as obtained in a model towing basin of dimensions 1 × 1 × 40 m. A recirculating region was found on both sides of the plume, but the secondary (and higher-order) cell structures proposed by other investigators were not observed. The primary cells were found to be appreciably longer than those reported in the literature for vertical plane jets. The difference can in part be attributed to different definitions of cell length. The definitions used herein are based on observables, both on the water surface and in the interior of the flow, and they lead to consistent measures of length. Bubble-plume parameters (such as entrainment coefficient) are known to depend on gas flow rate, and it was found that the length of the primary cell is a weak function of this variable as well. Additional experiments with a vertical plane jet were conducted for comparison. Longer cells than those previously reported were again observed. The paper contains a complete theory for line bubble plumes, including the effects of compressibility, bubble slip and finite release volume, as well as a simplified similarity analysis useful in estimating plume properties and horizontal-current depth and velocity.

Oceanic observations from western Europe and the south-western Indian ocean have provided evidence of the generation of internal solitary waves due to an internal tidal beam impinging on the pycnocline from below – a process referred to as ‘local generation’ (as opposed to the more direct generation over topography). Here we present the first direct numerical simulations of such a generation process with a fully nonlinear non-hydrostatic model for an idealised configuration. We show that, depending on the parameters, different modes can be excited and we provide examples of internal solitary waves as first, second and third modes, trapped in the pycnocline. A criterion for the selection of a particular mode is put forward, in terms of phase speeds. In addition, another simpler geometrical criterion is presented to explain the selection of modes in a more intuitive way. Finally, results are discussed and compared with the configuration of the Bay of Biscay.

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In this paper the translatory motion of a compound drop is examined in detail for low-Reynolds-number flow. The compound drop, consisting of a liquid drop or a gas bubble completely coated by another liquid, moves in a third immiscible fluid. An exact solution for the flow field is found in the limit of small capillary numbers by approximating the two interfaces to be spherical. The solution is found for the general case of eccentric configuration with motion of the inner sphere relative to the outer together with the motion of the system in the continuous phase. The results show that the viscous forces tend to move the inner-fluid sphere towards the front stagnation point of the compound drop. For equilibrium of the inner sphere with respect to the outer there must, therefore, be a body force towards the front. This can only be achieved with the necessary condition that there be a buoyant force on the inner sphere, opposite to that of the compound drop in the continuous phase. For a given set of fluids, two or four equilibrium configurations may be found. Of these only one or two, respectively, are stable. The others are unstable. For the special case of concentric configuration, the equilibrium is always metastable.

Although unsteady, high-Reynolds-number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary-layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady three-dimensional separating flows follow and depend on the symmetry properties of the flow (e.g. line symmetry, axial symmetry). In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi two-dimensional with a displacement thickness in the form of a crescent-shaped ridge. Physically the singularities can be understood in terms of the behaviour of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Stochastic bifurcations and stability of natural convection within two-dimensional square enclosures are investigated by different stochastic modelling approaches. Deterministic stability analysis is carried out first to obtain steady-state solutions and primary bifurcations. It is found that multiple stable steady states coexist, in agreement with recent results, within specific ranges of Rayleigh number. Stochastic simulations are then conducted around bifurcation points and transitional regimes. The influence of random initial flow states on the development of supercritical convection patterns is also investigated. It is found that a multi-element polynomial chaos method captures accurately the onset of convective instability as well as multiple convection patterns corresponding to random initial flow states.

Landscapes that are rhythmically dissected by natural drainage channels exist in various geologic and climatic settings. Such landscapes are characterized by a length-scale for the lateral spacing between channels. We observe a small-scale version of this process in the form of beach rills and reproduce channelization in a table-top seepage experiment. On the beach as well as in the experiments, channels are spontaneously incised by surface flow, but once initiated, they grow due to water emerging from underground. Field observation and experiment suggest the process can be described in terms of flow through a homogeneous porous medium with a freely shaped water table. According to this theory, small deformations of the underground water table amplify the flux into the channel and lead to further growth, a phenomenon we call ‘Wentworth instability’. Piracy of groundwater can occur over distances much larger than the channel width. Channel spacing coarsens with time, until channels reach their maximum length.

We present a detailed investigation of the parametric subharmonic resonance mechanism that leads a plane, monochromatic, small-amplitude internal gravity wave, also referred to as the primary wave, to instability. Resonant wave interaction theory is used to derive a simple kinematic model for the parametrically forced perturbation, and direct numerical simulations of the Boussinesq equations in a vertical plane permit the nonlinear simulation of the internal gravity wave field. The processes that eventually drive the wave field to breaking are also addressed.

We show that parametric instability may be viewed as an optimized scenario for drawing energy from the primary wave, that is, from a periodic flow with both oscillating shear and density gradient. Optimal energy exchange maximizing perturbation growth is realized when the perturbation has a definite spatio-temporal structure: its energy is phase-locked with the vorticity of the primary wave. This organization allows the perturbation energy to alternate between kinetic form when locally the primary wave shear is negative, then maximizing kinetic energy extraction from the primary wave, and potential form when the primary wave shear is positive, then minimizing the reverse transfer to that wave. The perturbation potential energy increases through the primary wave density gradient whether the latter is positive, that is when the medium is of reduced static stability, or negative (increased static stability). When the primary wave amplitude is small, all energy transfer terms are predicted well by the kinematic model. One important result is that the rate of potential energy transfer from the primary wave to the perturbation is always larger than the rate of kinetic energy transfer, whatever the primary wave.

As the perturbation amplifies, overturned isopycnals first appear in reduced static stability regions, implying that the total field should become unstable through a buoyancy induced (or Rayleigh–Taylor) instability. Hence, a two-dimensional model is no longer valid for studying the subsequent flow development.

A study is made of the wind-driven circulation of a two-layer ocean within a square basin, with a view to describing the observed separation of western boundary currents. The lower layer is allowed to surface and the line along which the upper-layer depth vanishes is interpreted as the region of the surfacing thermocline. For a representative wind stress the theory predicts the gross features of the Gulf Stream flow, the region adjacent to the surfacing line containing the separated boundary current. By assuming that the effects of friction and inertia are confined to regions of a boundary-layer character, the position of a separated current is shown to depend only on the degree of stratification and certain integral properties of the applied wind stress.

We consider the fluid flow in a spherical shell in rapid overall rotation, with additionally a differential rotation imposed on the inner sphere. The basic state consists of the axisymmetric Stewartson shear layer situated on the tangent cylinder, the cylinder parallel to the axis of rotation and just touching the inner sphere. In this work we consider the non-axisymmetric instabilities that arise when the differential rotation becomes sufficiently large. We find that the sign of the differential rotation, that is, whether the inner sphere is rotating slightly faster or slightly slower than the outer sphere, is crucial, with positive differential rotations yielding a progression to higher wavenumbers $m$ as the overall rotation rate increases, but negative differential rotations yielding $m\,{=}\,1$ over almost the entire range of rotation rates. This difference is particularly intriguing, as it has been seen before in one closely related experimental study, but not in another. A prior asymptotic analysis also suggested there should be no difference. We therefore try to understand what subtle features of the flow structures and/or geometries should cause this difference in results. We show that the geometry is the critical feature, with the height along the axis of rotation changing abruptly across the tangent cylinder. We are not able to identify why this should make such a difference, and why only for negative differential rotations. We suggest instead additional experiments and asymptotics to further clarify this point.

We develop a physical and computational model for performing fully coupled, grain-resolved direct numerical simulations of cohesive sediment, based on the immersed boundary method. The model distributes the cohesive forces over a thin shell surrounding each particle, thereby allowing for the spatial and temporal resolution of the cohesive forces during particle–particle interactions. The influence of the cohesive forces is captured by a single dimensionless parameter in the form of a cohesion number, which represents the ratio of cohesive and gravitational forces acting on a particle. We test and validate the cohesive force model for binary particle interactions in the drafting–kissing–tumbling (DKT) configuration. Cohesive sediment grains can remain attached to each other during the tumbling phase following the initial collision, thereby giving rise to the formation of flocs. The DKT simulations demonstrate that cohesive particle pairs settle in a preferred orientation, with particles of very different sizes preferentially aligning themselves in the vertical direction, so that the smaller particle is drafted in the wake of the larger one. This preferred orientation of cohesive particle pairs is found to remain influential for systems of higher complexity. To this end, we perform large simulations of 1261 polydisperse settling particles starting from rest. These simulations reproduce several earlier experimental observations by other authors, such as the accelerated settling of sand and silt particles due to particle bonding, the stratification of cohesive sediment deposits, and the consolidation process of the deposit. They identify three characteristic phases of the polydisperse settling process, viz. (i) initial stir-up phase with limited flocculation, (ii) enhanced settling phase characterized by increased flocculation, and (iii) consolidation phase. The simulations demonstrate that cohesive forces accelerate the overall settling process primarily because smaller grains attach to larger ones and settle in their wakes. For the present cohesive number values, we observe that settling can be accelerated by up to 29 %. We propose physically based parametrization of classical hindered settling functions introduced by earlier authors, in order to account for cohesive forces. An investigation of the energy budget shows that, even though the work of the collision forces is much smaller than that of the hydrodynamic drag forces, it can substantially modify the relevant energy conversion processes.

The distribution of pressure in normal and stenosed aortic valves is investigated experimentally with a rigid-walled model placed in a pulsatile water-tunnel, and the experiments are complemented by a one-dimensional solution of the unsteady inviscid-flow equations. In the normal valve, convectively fed vortices are formed in the aortic sinuses; the vortices aid cusp positioning and the prevention of jet formation during valve closure. Aortic valve stenosis is shown to prevent the generation of vortices, causing the formation of a turbulent jet, with reduction of the pressure difference between the inlets (ostia) of the coronary arteries and the ventricle. This pressure difference is calculated for man resting and exercising, and for various degrees of stenosis.