The feasibility of an experimental method for investigations of the particle flux to an absorbing surface in turbulent flows is demonstrated in a Lagrangian as well as an Eulerian representation. A laboratory experiment is carried out, where an approximately homogeneous and isotropic turbulent flow is generated by two moving grids. The simultaneous trajectories of many small approximately neutrally buoyant polystyrene particles are followed in time. In a Lagrangian analysis, we select one of these as the centre of a ‘sphere of interception’, and obtain estimates for the time variation of the statistical average of the inward particle flux through the surface of this moving sphere. The variation of the flux with the radius in the sphere of interception, as well as the variation with basic flow parameters is described well by a simple model, in particular for radii smaller than a characteristic length scale for the turbulence. The Eulerian counterpart of the problem is analysed as well, and the two results compared. Applications of the problem to, for instance, the question of the feeding rate of micro-organisms in turbulent marine environments are pointed out.

The development of a subharmonic three-dimensional instability mode in a vortex street is investigated both numerically and experimentally. The flow past a ring is considered as a test case, as a previous stability analysis has predicted that for a range of aspect ratios, the first-occurring instability of the vortex street is subharmonic. For the flow past a circular cylinder, the development of three-dimensional flow in the vortex street is known to lead to turbulent flow through the development of spatio-temporal chaos, whereas subharmonic instabilities have been shown to cause a route to chaos through the development of a period-doubling cascade. The three-dimensional vortex street in the flow past a ring is analysed to determine if a subharmonic instability can alter the route to turbulence for a vortex street.

A linear stability analysis and non-axisymmetric computations are employed to compute the flow past a ring with an aspect ratio ${\sc ar}\,{=}\,5$, and comparisons with experimental dye visualizations are included to verify the existence of a subharmonic mode in the wake. Computations at higher Reynolds numbers confirm that the subharmonic instability does not initiate a period-doubling cascade in the wake.

This paper assesses the potential of coherent vortex simulation (CVS) to compute three-dimensional turbulent mixing layers. CVS splits each turbulent flow realization into two orthogonal parts, one corresponding to coherent vortices which are kept, and the other to an incoherent background flow which is discarded. The CVS filter is applied to data from direct numerical simulations (DNS) of three-dimensional forced and unforced time-developing turbulent mixing layers. The coherent flow is represented by few wavelet modes, but these are sufficient to reproduce the vorticity probability distribution function and the energy spectrum out to the high-wavenumber end of the inertial range. The discarded incoherent background flow is homogeneous, small-amplitude and decorrelated. The CVS-filtering results are then compared with those obtained for the same compression ratio using Fourier low-pass filtering, as employed in large-eddy simulation (LES). Compared to the incoherent background flow of CVS filtering, the subgrid scales of LES filtering are less homogeneous, have much larger amplitude, and exhibit spatial correlations that makes modelling them a difficult challenge. Finally we present simulations of a time-developing mixing layer where the CVS filter is applied at each time step. The results show that CVS preserves the nonlinear dynamics of the flow, and that discarding the incoherent modes is sufficient to model turbulent dissipation.

New and unexpected results are presented regarding the nonlinear interactions between a wavepacket and a vortical mean flow, with an eye towards internal wave dynamics in the atmosphere and oceans and the problem of ‘missing forces’ in atmospheric gravity-wave parametrizations. The present results centre around a pre-wave-breaking scenario termed ‘wave capture’, which differs significantly from the standard such scenarios associated with critical layers or mean density decay with altitude. We focus on the peculiar wave–mean interactions that accompany wave capture. Examples of these interactions are presented for layerwise-two-dimensional, layerwise-non-divergent flows in a three-dimensional Boussinesq system, in the strong-stratification limit.

The nature of the interactions can be summarized in the phrase ‘wave–vortex duality’, whose key points are firstly that wavepackets behave in some respects like vortex pairs, as originally shown in the pioneering work of Bretherton (1969), and secondly that a collection of interacting wavepackets and vortices satisfies a conservation theorem for the sum of wave pseudomomentum and vortex impulse, provided that the impulse is defined appropriately. It must be defined as the rotated dipole moment of the Lagrangian-mean potential vorticity (PV). This PV differs crucially from the PV evaluated from the curl of either the Lagrangian-mean or the Eulerian-mean velocity. The results are established here in the strong-stratification limit for rotating (quasi-geostrophic) as well as for non-rotating systems. The concomitant momentum budgets can be expected to be relatively complicated, and to involve far-field recoil effects in the sense discussed in Bühler & McIntyre (2003). The results underline the three-way distinction between impulse, pseudomomentum, and momentum. While momentum involves the total velocity field, impulse and pseudomomentum involve, in different ways, only the vortical part of the velocity field.

The effect of particle–particle contact on the stress of a suspension of small spheres in plane strain flow is investigated. We provide an analytic form for the particle pair distribution function in the case of no Brownian motion, and calculate the viscosity and normal stress difference based on this. We show that the viscosity is reduced by contact, and a normal stress difference induced, both at order $c^2$ for small particle volume concentration $c$. In addition, we investigate the effect of a small amount of diffusion on the structure of the distribution function, giving a self-consistent form for the density in the $O(aPe^{-1})$ boundary layer and demonstrating that diffusion reduces the magnitude of the contact effect but does not qualitatively alter it.

This paper presents simulations of the two-dimensional model developed by Pothérat et al. (2000) for MHD flows between two planes with a strong transverse homogeneous and steady magnetic field, accounting for moderate inertial effects in Hartmann layers. We first show analytically how the additional terms in the equations of motion accounting for inertia soften velocity gradients in the horizontal plane, and then we implement the model in a code to carry out numerical simulations to be compared with available experimental results. This comparison shows that the new model can give very accurate results as long as the Hartmann layer remains laminar. Both experimental velocity profiles and global angular momentum measurements are recovered, and local and global Ekman recirculations are shown to alter significantly the shape of the flow as well as the global dissipation.

We study the linearized Navier–Stokes (LNS) equations in channel flows from an input–output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien–Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input–output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input–output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field.

Flow past a stationary cylinder becomes unstable at Re$\,{\sim}\,47$. Flow-induced vibrations of an elastically mounted cylinder, of low non-dimensional mass, is investigated at subcritical Reynolds numbers. A stabilized finite-element formulation is used to solve the incompressible flow equations and the cylinder motion in two dimensions. The cylinder is free to vibrate in both the transverse and in-line directions. It is found that, for certain natural frequencies of the spring–mass system, vortex shedding and self-excited vibrations of the cylinder are possible for Re as low as 20. Lock-in is observed in all cases. However, the mass of the oscillator plays a major role in determining the proximity of the vortex-shedding frequency to the natural frequency of the oscillator. A global linear stability analysis (LSA) for the combined flow and oscillator is carried out. The results from the LSA are in good agreement with the two-dimensional direct numerical simulations.

The convergence rate and the structures of the proper orthogonal decomposition (POD) reconstruction are re-examined. The relatively slow convergence rate of the wall-normal velocity and the over-prediction of the Reynolds shear stress from POD-based representations of near-wall turbulence suggest that coherent structures are too well correlated. This is a consequence of the directional preference of the eigenfunctions toward the most energetic data (the streamwise motion at $x^+_2 \,{=}\, 13$) which may be relaxed by extracting the POD eigenfunctions from the streak modes and the streamwise vortex modes separately. The rate of convergence and representation of structures may be improved by performing the POD in Squire's coordinate system (PODS). The statistical reconstruction of the localized ($x^+_2 \,{\leq}\, 84$) PODS eigen-functions from the large-eddy simulation (LES) database suggests that the convergence rate of component energy, Reynolds shear stress and the correlation coefficient of streamwise and wall-normal velocity improve significantly. The PODS eigenvalue spectra and the spatial structures of the eigenfunctions indicate the presence of observed streaks, streamwise vortices, and ejection/sweep events.

Dynamical models of the full channel turbulence are constructed by projecting the Navier–Stokes equations onto the PODS eigenfunctions. The unresolved modes are taken into account by closure models adapted from a constant eddy-viscosity model, dynamic Smagorinsky, and dynamic one-coefficient mixed subgrid-scale (SGS) model. Statistical moments obtained from PODS-based simulations are much more accurate than their counterparts from POD-based simulations. This is particularly true for the mean velocity, streamwise turbulent intensity, Reynolds shear stress and the correlation coefficient of streamwise and wall-normal velocity.

A new law for the thinning of surfactant-free lamellae (applicable to metallic and ceramic foams with mobile interfaces) in a cross-section of an arid gas–liquid foam is derived using matched asymptotic analysis. Two limiting cases are identified at small capillary number: the well-known semi-arid foam having unit-order liquid fraction and the arid foam in which it is small. The lamellar thinning rates in both cases exhibit $t^{-2}$ power-law behaviour at long times even though the foam liquid area fractions have different orders of magnitude in capillary number. At early times, arid foam thinning is slowed because the curvature of the capillary quasi-static interfacial region must decrease in order to accommodate the flow from the films. Therefore, the thinning of lamellae feeding into a given Plateau border is coupled and the dynamics is distinct from that of the semi-arid foam.

Approximations of rupture times in arid and semi-arid foams are found by calculating the times for lamellae to thin to a pre-specified thickness. For given initial lamellar thicknesses, and for arid and semi-arid foams that have identical initial lamellar liquid areas, the arid foam ruptures more quickly than the semi-arid foam. On the other hand the rupture of lamellae is significantly delayed in arid foam compared to semi-arid foam if the initial lamellar thickness and capillary number are the same.

Measurements of the flow field around a model rotor descending axially into its own vortex wake have been performed using particle image velocimetry (PIV). At low descent rates, the expected cylindrical down-flow structure below the rotor is observed. At slightly higher descent rate, the flow enters the so-called vortex ring state (VRS) where the vorticity from the rotor accumulates into a toroidal structure near the rotor tips, and a large recirculation zone forms above the rotor disk. In the VRS, the flow below the rotor shows a significant upwards component, with a small up-flow zone penetrating right up to the rotor disk. Measurements show there to be a range of descent rates just before the onset of the VRS over which the flow may be interpreted to be in an incipient VRS condition. In this range, analyses of individual PIV measurements indicate that the flow near the rotor intermittently switches between the down-flow topology found at lower descent rates and the flow topology found in the fully developed VRS. The frequency of excursions of the flow into the VRS topology increases as the descent rate of the rotor is increased until, at high enough descent rate, the flow remains locked within its toroidal state.

In applications involving the model-based control of transitional wall-bounded flow systems, it is often desired to estimate the interior flow state based on a history of noisy measurements from an array of flush-mounted skin-friction and pressure sensors on the wall. This paper considers this estimation problem, using a Kalman filter based on the linearized Navier–Stokes equations and appropriate stochastic models for the relevant statistics of the initial conditions, sensor noise and external disturbances acting on the system. We show that a physically relevant parameterization of these statistics is key to obtaining well-resolved feedback kernels with appropriate spatial extent for all three types of flow measurement available on the wall. The effectiveness of the resulting Kalman and extended Kalman filters that implement this feedback is verified for both infinitesimal and finite-amplitude disturbances in direct numerical simulations of a perturbed laminar channel flow. The consideration of time-varying feedback kernels is shown to be particularly advantageous in accelerating the convergence of the estimator from unknown initial conditions. A companion paper (Part 2) considers the extension of such estimators to the case of fully developed turbulence.

A detailed experimental study on the formation of crossflow vortex mode packets and their high-frequency secondary instability in a swept-wing boundary layer was carried out. Stationary vortex packets are most likely to be generated under natural flight conditions and transition to turbulence is quickest within these disturbances. In the present experiments, different methods of controlled excitation are used so that the crossflow vortex packets are generated by surface-roughness elements and by localized continuous suction. It is found that as the stationary disturbance reaches a significant amplitude, of about 10% of the free-stream velocity, while being below the saturation level, high-frequency secondary instabilities start to grow. Influence of the crossflow vortex packet magnitude on the development of the secondary instability is investigated in detail and below its threshold the crossflow vortex packet was found to be nearly neutrally stable. By studying the unstable packets, the frequency of natural secondary perturbations was identified and the travelling disturbances were forced in a controlled manner by periodic blowing–suction applied locally under the stationary vortex. Two modes of secondary instability were found to develop and the preferred mode was dependent on the properties of the primary stationary disturbance. Additionally, the underlying physics of the process of nonlinear formation and development of the vortices in the boundary layer is clarified. It was observed that the large-amplitude co-rotating vortices may interact, thus reducing their amplitude. Also a large-scale excitation by an isolated roughness element produced two individual stationary crossflow vortex packets at its tips, each with different preferred secondary instability modes.

The relaminarization phenomenon in the laminar–turbulent transition process of spherical Couette flow with the inner sphere rotating and the outer sphere fixed has been experimentally investigated for $0.06\,{ \leqslant}\,\beta\,{\leqslant}\,0.206$, where $\beta$ is the ratio of the clearance to the inner-sphere radius. The relaminarization occurs for $0.13\,{<}\,\beta\,{<}\,0.17$, and is observed as a reverse Hopf bifurcation from the limit cycle to the fixed point. The kinetic energy in the high-frequency region of the fluctuating azimuthal velocity component continues to increase with Reynolds number Re in the case without relaminarization, but in the case with relaminarization it first increases and then decreases with increasing Re and finally vanishes with the onset of the relaminarization. For $\beta\,{=}\,0.14$ with relaminarization, a small artificial disturbance introduced externally into the spherical Couette flow has no influence on the lowest critical Reynolds number of the first instability that can be defined universally and uniquely. On the other hand, the external disturbance decreases the onset Reynolds numbers of both the second instability (occurrence of spiral Taylor–Görtler (TG) vortices) and the relaminarization, although it has no influence on the fundamental frequencies of velocity fluctuation caused by spiral TG vortices and travelling waves on TG vortices.

The flow around a vertical circular pile exposed to a steady current is studied numerically and experimentally. The numerical model is a three-dimensional model. The model validation was achieved against new experimental data (which include two-component laser-Doppler anemometry (LDA) flow measurements and the hot-film bed shear stress measurements, and reported in the present paper) and the data of others, and a $k$–$\omega $ turbulence model was used for closure. The model does not have a free-surface facility and therefore is applicable only to cases where the Froude number is small ($\hbox{\it Fr}\,{<}\,O(0.2)$). The flow model was used to study the horseshoe vortex and lee-wake vortex flow processes around the pile. The influence on the horseshoe vortex of three parameters, namely the boundary-layer thickness, the Reynolds number and the bed roughness, was investigated. In the latter investigation, the steady solution of the model was chosen. A study of the influence of the unsteady solution on the previously mentioned flow processes was also carried out. The ranges of the parameters covered in the numerical simulations are: The boundary-layer-thickness-to-pile-diameter ratio is varied from 2$\,{\times}\, 10^{-2}$ to 10$^{2}$, the pile Reynolds number from 10$^{2}$ to $2\,{\times}\, 10^{6},$ and the pile diameter-to-roughness ratio from 2 to about 10$^{3}.$ The amplification of the bed shear stress around the pile (including the areas under the horseshoe vortex and the lee-wake region) was obtained for various values of the previously mentioned parameters. The steady-state flow model was coupled with a morphologic model to calculate scour around a vertical circular pile exposed to a steady current in the case of non-cohesive sediment. The morphologic model includes (i) a two-dimensional bed load sediment-transport description, and (ii) a description of surface-layer sand slides for bed slopes exceeding the angle of repose. The results show that the present numerical simulation captures all the main features of the scour process. The equilibrium scour depth obtained from the simulation agrees well with the experiments for the upstream scour hole. Some discrepancy (up to 30%) was observed, however, for the downstream scour hole. The calculations show that the amplification of the bed shear stress around the pile in the equilibrium state of the scour process is reduced considerably with respect to that experienced at the initial stage where the bed is plane.

We consider an experiment in which a rectangular flat plate is flapped with two degrees of freedom, and a genetic algorithm tunes its trajectory parameters so as to achieve maximum average lift force, thus evolving a population of trajectories all yielding optimal lift forces. We cluster the converged population by defining a dynamical formation number for a flapping flat plate, thus showing that optimal unsteady force generation is linked to the formation of a leading-edge vortex with maximum circulation. Force and digital particle image velocimetry measurements confirm this result.