We employ a three-dimensional, nonlinear inviscid numerical method, in conjunction with experimental data from live fish and from a fish-like robotic mechanism, to establish the three-dimensional features of the flow around a fish-like body swimming in a straight line, and to identify the principal mechanisms of vorticity control employed in fish-like swimming. The computations contain no structural model for the fish and hence no recoil correction. First, we show the near-body flow structure produced by the travelling-wave undulations of the bodies of a tuna and a giant danio. As revealed in cross-sectional planes, for tuna the flow contains dominant features resembling the flow around a two-dimensional oscillating plate over most of the length of the fish body. For the giant danio, on the other hand, a mixed longitudinal–transverse structure appears along the hind part of the body. We also investigate the interaction of the body-generated vortices with the oscillating caudal fin and with tail-generated vorticity. Two distinct vorticity interaction modes are identified: the first mode results in high thrust and is generated by constructive pairing of body-generated vorticity with same-sign tail-generated vorticity, resulting in the formation of a strong thrust wake; the second corresponds to high propulsive efficiency and is generated by destructive pairing of body-generated vorticity with opposite-sign tail-generated vorticity, resulting in the formation of a weak thrust wake.

Mixing enhancement in a mixing layer is considered in terms of a ‘vortex generator’ that uses fluid dynamically generated counter-rotating longitudinal vortices rather than explicit winglets or similar devices. This view is reached through considering the centrifugal instability of weak initial Görtler vortices on a slightly concave wall that are allowed to develop to their various nonlinear stages through selecting the cutoff lengths of the trailing edge prior to their release into the mixing region. These vortices are released from one side of the (say, upper) stream in the present work. The quantitative entrainment properties of the longitudinal vortices are studied to select an optimal trailing-edge cutoff for fixed upstream conditions. As the vortices develop along the wall, they are intensified because of the centrifugal instability mechanism and because of the work done by the Reynolds stress of the vortices against the local mean flow rate of strain; simultaneously, the region of strong streamwise vorticity moves away from the wall. This selection process is explained through a balance between the vorticity strength and proximity to the lower stream when the trailing edge is cut off: it is shown, therefore, that vortices of relatively modest strength and kinetic energy that are close to the interface separating the two streams provide mixing properties superior to stronger vortices located too far from the interface. Energy-balancing mechanisms and the stretching of the initial interface are studied, as are the effects of the velocity ratio and the spanwise wavelengths other than the fundamental. In order further to enhance mixing by exploiting the inherent secondary instability of primary steady longitudinal vortices, the most amplified secondary instability of the optimal-trailing-edge cutoff situation, which is the sinuous mode, is studied in detail in terms of the nonlinear development and modification of the steady vortical flow. Local energy-exchange mechanisms are studied, as are the mixing properties of the modified steady flow, which are shown to be significantly improved compared to the unmodified steady flow. Though the initiation of steady longitudinal vortices relies on centrifugal instability upstream, such vortices are able to develop self-sustaining and amplifying properties through the Reynolds stresses in the mixing region even without centrifugal instability reinforcement. The secondary instability is initiated and sustained entirely through its own three-dimensional Reynolds stress properties, which work against the three-dimensional rates of strain in the entire steady flow. This contrasts with initially generated potential-like vortices that decay downstream in the presence of dissipative mechanisms without the production mechanisms due to the Reynolds stresses.

The behaviour of heavy particles in isotropic, homogeneous, decaying turbulence has been experimentally studied. The settling velocity of the particles has been found to be much larger than in a quiescent fluid. It has been determined that the enhancement of the settling velocity depends on the particle loading, increasing as the volume fraction of particles in the flow increases. The spatial and temporal distribution of the particle concentration field is shown to exhibit large inhomogeneities. As the particles interact with the underlying turbulence they concentrate preferentially in certain regions of the flow. A characteristic dimension of these particle clusters is found to be related to the viscous scales of the flow. Measurements of the settling velocity conditioned on the local concentration of particles in the flow have shown that there is a monotonic increase in the settling velocity with the local concentration (the relation being quasi-linear). A simple phenomenological model is proposed to explain this behaviour.

Experiments were performed to measure the rebound velocities of small plastic and metal spheres dropped from various heights onto a smooth quartz surface coated with a thin layer of viscous fluid. The spheres stick without rebounding for low impact velocities, due to viscous dissipation in the thin fluid layer. Above a critical impact velocity, however, the lubrication forces in the thin layer cause elastic deformation and rebound of the spheres. The apparent coefficient of restitution increases with the ratio of the Stokes number to its critical value for rebound, where the Stokes number is a dimensionless ratio of the inertia of the sphere to viscous forces in the fluid. The critical Stokes number required for rebound decreases weakly with increasing values of a dimensionless elasticity parameter which is a ratio of the viscous forces which cause deformation to the elastic forces which resist deformation. The experimental results show good agreement with an approximate model based on lubrication theory for undeformed spheres and scaling relations for elastic deformation.

We consider a class of three-dimensional boundary-layer flows, which may be viewed as an extension of the Falkner–Skan similarity form, to include a cross-flow velocity component, about a plane of symmetry. In general, this provides a range of three- dimensional boundary-layer solutions, parameterized by a Falkner–Skan similarity parameter, n, together with a further parameter, Ψ∞, which is associated with a cross-flow velocity component in the external flow. In this work two particular cases are of special interest: for n = 0 the similarity equations possess a family of solutions related to the Blasius boundary layer; for n = 1 the similarity solution provides an exact reduction of the Navier–Stokes equations corresponding to the flow near a saddle point of attachment. It is known from the work of Davey (1961) that in this latter class of flow, a continuum of solutions can be found. The continuum arises (in general) because it is possible to find states with an algebraic, rather than exponential, behaviour in the far field. In this work we provide a detailed overview of the continuum states, and show that a discrete infinity of ‘exponential modes’ are smoothly embedded within the ‘algebraic modes’ of the continuum. At a critical value of the cross-flow, these exponential modes appear as a cascade of eigensolutions to the far-field equations, which arise in a manner analogous to the energy eigenstates found in quantum mechanical problems described by the Schrödinger equation.

The presence of a discrete infinity of exponential modes is shown to be a generic property of the similarity equations derived for a general n. Furthermore, we show that there may also exist non-uniqueness of the continuum; that is, more than one continuum of states can exist, that are isolated for fixed n and Ψ∞, but which are connected through an unfolded transcritical bifurcation at a critical value of the cross-flow parameter, Ψ∞.

The multiplicity of states raises the question of solution selection, which is addressed using two stability analyses that assume the same basic symmetry properties as the base flow. In one case we consider a steady, algebraic form in the ‘streamwise’ direction, whilst in the other a temporal form is assumed. In both cases it is possible to extend the analysis to consider a continuous spectrum of disturbances that decay algebraically in the wall-normal direction. We note some obvious parallels that exist between such stability analyses and the approach to the continua of states described earlier in the paper.

We also discuss the appearance of analogous non-unique states to the Falkner–Skan equation in the presence of an adverse pressure gradient (i.e. n < 0) in an appendix.

A magnetohydrodynamic (MHD) stirrer that exhibits chaotic advection is investigated experimentally and theoretically. The stirrer consists of a circular cavity with an electrode (C) deposited around its periphery. Two additional electrodes (A) and (B) are deposited eccentrically inside the cavity on the bottom. The cavity is positioned in a uniform magnetic field that is parallel to the cylinder's axis, and it is filled with a weak electrolyte solution. Fluid motion is induced in the cavity by applying a potential difference across a pair of electrodes. A closed-form, analytical solution is derived for the MHD creeping flow field in the gap between the two eccentric cylinders. A singular solution is obtained for the special case when the size of the inner electrode shrinks to a point. Subsequently, passive tracers' trajectories are computed when the electric potential differences are applied alternately across electrodes AC and BC with period T. At small periods T, the flow is regular and periodic in most of the cavity. As the period increases, so does the complexity of the motion. At relatively large periods, the passive tracer experiences global chaotic advection. Such a device can serve as an efficient stirrer. Since this device has no moving parts, it is especially suitable for microfluidic applications. This is yet another practical example of a modulated, two-dimensional Stokes flow that exhibits chaotic advection.

A similarity solution to the long-wave shallow-water equations is obtained for a density current (reduced gravity = g′, Coriolis parameter = f) propagating alongshore (y = 0). The potential vorticity q = f/H1 is uniform in −∞ < x [les ] xnose(t), 0 < y [les ] L(x, t), and the nose of this advancing potential vorticity front displaces fluid of greater q = f/H0, which is located at L < y < ∞. If L0 = L(−∞, t), the nose point with L(xnose(t), t) = 0 moves with velocity Unose = √g′H0 φ, where φ is a function of H1/H0, f2L20/g′H0. The assumptions made in the similarity theory are verified by an initial value solution of the complete reduced-gravity shallow-water equations. The latter also reveal the new effect of a Kelvin shock wave colliding with a potential vorticity front, as is confirmed by a laboratory experiment. Also confirmed is the expansion wave structure of the intrusion, but the observed values of Unose are only in qualitative agreement; the difference is attributed to the presence of small-scale (non-hydrostatic) turbulence in the laboratory experiment but not in the numerical solutions.

We have simulated the dynamics of suspensions of fibres sedimenting in the limit of zero Reynolds number. In these simulations, the dominant inter-particle force arises from hydrodynamic interactions between the rigid, non-Brownian fibres. The simulation algorithm uses slender-body theory to model the linear and rotational velocities of each fibre. To include far-field interactions between the fibres, the line distribution of force on each fibre is approximated by making a Legendre polynomial expansion of the disturbance velocity on the fibre, where only the first two terms of the expansion are retained in the calculation. Thus, the resulting linear force distribution can be specified completely by a centre-of-mass force, a couple, and a stresslet. Short-range interactions between particles are included using a lubrication approximation, and an infinite suspension is simulated by using periodic boundary conditions. Our numerical results confirm that the sedimentation of these non-spherical, orientable particles differs qualitatively from the sedimentation of spherical particles. The simulations demonstrate that an initially homogeneous, settling suspension develops clusters, or streamers, which are particle rich surrounded by clarified fluid. The instability which causes the heterogeneous structure arises solely from hydrodynamic interactions which couple the particle orientation and the sedimentation rate in particle clusters. Depending upon the concentration and aspect ratio, the formation of clusters of particles can enhance the sedimentation rate of the suspension to a value in excess of the maximum settling speed of an isolated particle. The suspension of fibres tends to orient with gravity during the sedimentation process. The average velocities and orientations, as well as their distributions, compare favourably with previous experimental measurements.

The objective of this work is a numerical study of the stability properties and the evolution of the eastward-travelling baroclinic modons – coherent vortex structures specific to stratified geophysical fluids where differential rotation (the β-effect) is of the essence. In the vortices under study, the initial dependence of the potential vorticity (PV) upon the streamfunction is piecewise-linear, the barotropic component is dipolar, the baroclinic component is circularly symmetric about the vertical axis, and the boundary of the trapped-fluid region (in which the vorticity contours are closed) is a circular cylinder. These modons are shown to be stable for a wide range of parameters. In two- and three-layer fluids, modons of this type are shown to be able to transit to even more durable states, in which the trapped-fluid area is oval in shape and the PV versus streamfunction dependence in this domain is nonlinear. Possible transition mechanisms and linkage between the circular and oval modons are discussed.

While turbulent drag reduction through the injection of micro-bubbles into a turbulent boundary layer is well established in experiments, there is a lack of corresponding supporting evidence from direct numerical simulations. Here we report on a series of numerical simulations of small bubbles seeded in a turbulent channel flow at average volume fractions of up to 8%. These results show that even for relatively large bubbles, an initial transient drag reduction can occur as bubbles disperse into the flow. Relatively small spherical bubbles will produce a sustained level of drag reduction over time.

Particle transfer in the wall region of turbulent boundary layers is dominated by the coherent structures which control the turbulence regeneration cycle. Coherent structures bring particles toward and away from the wall and favour particle segregation in the viscous region, giving rise to non-uniform particle distribution profiles which peak close to the wall. The object of this work is to understand the reasons for higher particle concentration in the wall region by examining turbulent transfer of heavy particles to and away from the wall in connection with the coherent structures of the boundary layer. We will examine the behaviour of a dilute dispersion of heavy particles – flyashes in air – in a vertical channel flow, using pseudo-spectral direct numerical simulation to calculate the turbulent flow field at a shear Reynolds number Reτ = 150, and Lagrangian tracking to describe the dynamics of particles. Drag force, gravity and Saffman lift are used in the equation of motion for the particles, which are assumed to have no influence on the flow field. Particle interaction with the wall is fully elastic. As reported in several previous investigations, we found that particles are transferred by sweeps – Q2 type events – in the wall region, where they preferentially accumulate in the low-speed streak environments, whereas ejections – Q4 type events – transfer particles from the wall region to the outer flow. We quantify the efficiency of the instantaneous realizations of the Reynolds stresses events in transferring different size particles to the wall and away from the wall, respectively. Our findings confirm that sweeps and ejections are efficient transfer mechanisms for particles. In particular, we find that only those sweep and ejection events with substantial spatial coherence are effective in transferring particles. However, the efficiency of the transfer mechanisms is conditioned by the presence of particles to be transferred. In the case of ejections, particles are more rarely available since, when in the viscous wall layer, they are concentrated under the low-speed streaks. Even though the low-speed streaks are ejection-like environments, particles remain trapped for a long time. This phenomenon, which causes accumulation of particles in the near-wall region, can be interpreted in terms of overall fluxes toward and away from the wall by the theory of turbophoresis. This theory, proposed initially by Caporaloni et al. (1975) and re-examined later by Reeks (1983), can help to explain the existence of net particle fluxes toward the wall as a manifestation of the skewness in the velocity distribution of the particles (Reeks 1983). To understand the local and instantaneous mechanisms which give rise to the phenomenon of turbophoresis, we focus on the near-wall region of the turbulent boundary layer. We examine the role of the rear-end of a quasistreamwise vortex very near to the wall in preventing particles in the proximity of the wall from being re-entrained by the pumping action of the large, farther from the wall, forward-end of a following quasi-streamwise vortex. We examine several mechanisms for turbulence structures near the wall and we find that the mechanism based on the archetypal quasi-streamwise structures identified by Schoppa & Hussain (1997), the parent–offspring regeneration cycle for near-wall quasi-streamwise vortices discussed by Brooke & Hanratty (1993), and the mechanism based on coherent packets of hairpin vortices, the fundamental super-structure characterized by Adrian, Meinhart & Tomkins (2000), all depict the same characteristic pattern which is responsible for particle trapping very near to the wall.

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

The lift and drag forces on an isolated particle resulting from an oscillating wall- bounded flow, are approximated using direct numerical simulation and extrapolation techniques. We also confirm the existence of anomalies in the lift force, which arise from the interaction of the vortical field with the particle. Anomalies can also occur for computational reasons and these are discussed as well.

This study was motivated by a long-standing question about the importance of lift forces in the dynamics of sediments in oceanic settings. To answer this question we use the numerically generated data as well as extrapolations to compute the ratio of the lift to buoyancy forces on a particle. This analysis suggests that for particles and oceanic conditions typical of the nearshore, the lift force can play a role in the dynamics of sedimentary beds.