Scalars with different molecular diffusivities can be transported at different rates in a strongly stratified, weakly turbulent flow. Rapid distortion theory (RDT) is used to examine the mechanisms responsible for differential diffusion of scalars in a sheared stratified flow. The theory, which applies when the flow is strongly stratified, predicts upgradient flux and its wavenumber dependence, which previous direct numerical simulations have shown to be important in differential diffusion. The net effect of shear on differential diffusion depends on the Grashof number, or the relative importance of buoyancy and viscous effects. RDT also allows the effects of the density ratio, Schmidt number, Lewis number, scalar activity and mean shear to be examined without the high computational cost of direct numerical simulation. RDT predicts that differential diffusion will increase with increasing density ratio, but only at low Grashof number. When the Lewis number is fixed, the Grashof number below which differential diffusion occurs decreases with increasing Schmidt number, and when one of the Schmidt numbers is fixed, differential diffusion decreases with increasing Lewis number. Also, differential transport of passive scalars increases when the Schmidt number of the scalar stratifying the flow increases.

Direct numerical simulation (DNS) is performed to investigate the fully developed turbulence in a straight square annular duct. The mean flow field and the turbulent statistics are compared with existing experiments and numerical results. The comparisons and the analysis of the DNS data led to the discovery of the turbulent boundary layers of concave and convex 90° corners, a corner flow similarity and the scaling characteristics of corner turbulence. Analysis of the mean streamwise velocity near the concave and convex 90° corners resulted in establishing the ‘law-of-the-corner’ formulations. Comparing these formulations with the ‘law-of-the-wall’ relation, both damping and enhancing mechanisms analytically represented by the van Driest damping function, and the enhancement function were revealed for the concave and convex corner turbulence. The investigation captures the distinctive turbulence-driven secondary flows for both convex and concave 90° corners, and a corner flow similarity rule is discovered, which is associated with the pattern of these secondary flows. A turbulence energy spectrum analysis provides the distinctive features of the fully developed turbulence in the wall and corner regions. The validity of the turbulence eddy viscosity concept is evaluated based on these turbulence energy spectra. The turbulence-driven secondary-flow generation mechanisms are investigated by analysing the anisotropy of the Reynolds stresses.

We present a prediction of the lateral force exerted on a rigid neutrally buoyant sphere in circular cross-section Poiseuille flow. The force is calculated with the method of matched asymptotic expansions. We investigate the influence of the pipe Reynolds number in the range 1–2000 on the equilibrium position and the magnitude of the lateral force. We show that the predicted lift force in a circular geometry is qualitatively similar to, but quantitatively different from, that in a plane channel. The predicted force in the pipe is significantly smaller than the channel result, and the zero of the force which determines the equilibrium radial position of a suspended particle lies closer to the centreline in the pipe.

We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter=9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 108. Under these conditions the convection is turbulent, and the time-averaged velocity at any point is ‘zero’. The Reynolds number based on the Taylor microscale, Reλ, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6–7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as Ra1/2Sc1/2, and the Reynolds number would scale as Ra1/2Sc−1/2. The velocity and the flux measurements appear to be consistent with the Ra1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are much higher compared to what would be obtained in Rayleigh–Bénard (R–B) convection for similar density differences.

The uniform flow past a sphere undergoing steady rotation about an axis transverse to the free stream flow was investigated numerically. The objective was to reveal the effect of sphere rotation on the characteristics of the vortical wake structure and on the forces exerted on the sphere. This was achieved by solving the time-dependent, incompressible Navier–Stokes equations, using an accurate Fourier–Chebyshev spectral collocation method. Reynolds numbers Re of 100, 250 and 300 were considered, which for a stationary sphere cover the axisymmetric steady, non-axisymmetric steady and vortex shedding regimes. The study identified wake transitions that occur over the range of non-dimensional rotational speeds Ω* = 0 to 1.00, where Ω* is the maximum velocity on the sphere surface normalized by the free stream velocity. At Re = 100, sphere rotation triggers a transition to a steady double-threaded structure. At Re = 250, the wake undergoes a transition to vortex shedding for Ω* ≥ 0.08. With an increasing rotation rate, the recirculating region is progressively reduced until a further transition to a steady double-threaded wake structure for Ω* ≥ 0.30. At Re = 300, wake shedding is suppressed for Ω* ≥ 0.50 via the same mechanism found at Re = 250. For Ω* ≥ 0.80, the wake undergoes a further transition to vortex shedding, through what appears to be a shear layer instability of the Kelvin–Helmholtz type.

Experiments have established that an axisymmetric sharp trailing edge of a gas bubble in a viscoelastic fluid can develop into a three-dimensional ‘knifelike’ shape under certain conditions (high capillary number, large bubble size). A numerical study is conducted to discover the physics of this phenomenon. The axisymmetric deformation of a bubble rising buoyantly in a viscoelastic fluid is simulated by solving the axisymmetric flow equations coupled with the constitutive equations of the finitely extensible nonlinear elastic Chilcott–Rallison (FENE-CR) model. The three-dimensional temporal linear stability analysis of this axisymmetric base state is carried out. The dominant eigenvalue which is indicative of the growth rate of the perturbations is computed. The only unstable eigenmode has azimuthal wavenumber m equal to 2. The corresponding eigenfunction shows that indeed a sharp axisymmetric tail develops a knife-edge form. A further investigation of the energy budget of the disturbances for m = 2 is performed to determine the production and dissipation terms affecting the growth of this instability. It is shown that the normal gradient of the base-state pressure along the free surface plays an important role in the evolution of the instability.

We consider the evolution of slender viscous fibres with cross-section containing holes with application to fabrication of microstructured optical fibres. The fibre evolution is driven by either prescribing velocity or a force at the ends of the fibre, and the free surfaces evolve under the influence of surface tension, internal pressurization, inertia and gravity. We use the fact that ratio of the typical fibre radius to the typical fibre length is small to perform an asymptotic analysis of the full three-dimensional Navier–Stokes equations similar to earlier work on non-axisymmetric (but simply connected) fibres. A numerical solution to the multiply connected steady-state drawing problem is formulated based on the solution the Sherman–Lauricella equation. The effects of different drawing and material parameters like surface tension, gravity, inertia and internal pressurization on the drawing are examined, and extension of the method to non-isothermal evolution is presented.

We introduce a model representing the venous network of the leg. The network consists of a coupled system of elastic tubes. The flow through each elastic tube is assumed to be unsteady, incompressible and one-dimensional. The network topology, as well as the lengths and diameters of the tubes, is based on literature data. As in the human leg the network is composed of two sub-networks, deep and superficial, which are connected by transverse segments. We introduce a new model for confluences or branching points, as well as models of the valvular system and of the muscular activity. We perform a numerical study of the transmission and reflection of waves at a confluence. Our model valvular system imposes a privileged direction of the flow towards the heart. Muscular activity is modelled using a modification of the tube law of the vessel and through an inflow of blood when muscle contraction pushes blood from the microcirculation to the veins. The model is capable of simulating several motions such as walking, dorsal flexion and tiptoe. Numerical tests show the physical relevance of the model, and in particular demonstrate that when the system is excited at the foot level, a two-frequency response appears. These frequencies are closely related to the characteristic lengths of the typical segments of the deep and of the superficial networks. We find good qualitative agreement between experimental and numerical flow rates, using clinical data corresponding to a single ‘tiptoe’ motion. We make numerical predictions of the internal venous pressure at the foot level in a valvular-incontinent system which agree with clinical observations.

In order better to understand how dynamo systems saturate, we study the kinematic dynamo properties of velocity fields that arise from nonlinearly saturated dynamos. The technique is implemented by solving concurrently, in addition to the momentum equation, two induction equations, one for the actual magnetic field and one for an independent passive vector field. We apply this technique to two illustrative examples: convectively driven turbulence and turbulence represented by a shell model. In all cases we find that the velocity remains an efficient kinematic dynamo even after nonlinear saturation occurs. We discuss the implications to the process of dynamo saturation.

We calculate the nonlinear response of an infinite ice sheet to a moving load in the time domain in two dimensions, using a higher-order spectral method. The nonlinearity is due to the moving boundary, as well as the nonlinear term in Bernoulli's equation and the elastic plate equation. We compare the nonlinear solution with the linear solution and with the nonlinear solution found by Parau & Dias (J. Fluid Mech., vol. 460, 2002, pp. 281–305). We find good agreement with both solutions (with the correction of an error in the Parau & Dias 2002 results) in the appropriate regimes. We also derive a solitary wavelike expression for the linear solution – close to but below the critical speed at which the phase speed has a minimum. Our model is carefully validated and used to investigate nonlinear effects. We focus in detail on the solution at a critical speed at which the linear response is infinite, and we show that the nonlinear solution remains bounded. We also establish that the inclusion of nonlinearities leads to significant new behaviour, which is not observed in the linear solution.

The steady flow of a stream emerging from a nozzle, hitting a horizontal plate and falling under gravity is considered. Depending on the length of the plate L and the Froude number F, the plate can either divert the stream or lead to its detachment. First, the problem is reformulated using conformal mappings. The resulting problem is then solved by a collocation Galerkin method; a particular form is assumed for the solution, and certain coefficients in that representation are then found numerically by satisfying Bernoulli's equation on the free surfaces at certain discrete points. The resulting equations are solved by Newton's method, yielding various configurations of the solution based on the values of F and L. The lift exerted on the plate is computed and discussed. If the plate is long enough, physically meaningful solutions are found to exist only for values of F greater than or equal to a certain critical value F0, which is to be determined. Results are presented, both for F > F0 where the detachment is horizontal and for F = F0 where the detachment point is a stagnation point at a 120° corner. Related asymmetric flows where the rising stream is inclined are also studied.

A set of controlled high-Reynolds-number experiments has been conducted at the William B. Morgan Large Cavitation Channel (LCC) in Memphis, Tennessee to investigate the friction drag reduction achieved by injecting aqueous poly(ethylene oxide) (PEO) solutions at three different mean molecular weights into the near-zero-pressure-gradient turbulent boundary layer that forms on a smooth flat test surface having a length of nearly 11m. The test model spanned the 3.05m width of the LCC test section and had an overall length of 12.9m. Skin-friction drag was measured with six floating-plate force balances at downstream-distance-based Reynolds numbers as high as 220 million and free stream speeds up to 20ms−1. For a given polymer type, the level of drag reduction was measured for a range of free stream speeds, polymer injection rates and concentrations of the injected solution. Polymer concentration fields in the near-wall region (0 < y+ < ~103) were examined at three locations downstream of the injector using near-wall planar laser-induced-fluorescence imaging. The development and extent of drag reduction and polymer mixing are compared to previously reported results using the traditional K-factor scaling. Unlike smaller scale and lower speed experiments, speed dependence is observed in the K-scaled results for the higher molecular weight polymers and it is postulated that this dependence is caused by molecular aggregation and/or flow-induced polymer degradation (chain scission). The evolution of near-wall polymer concentration is divided into three regimes: (i) the development region near the injector where drag reduction increases with downstream distance and the polymer is highly inhomogeneous forming filaments near the wall, (ii) the transitional mixing region where drag reduction starts to decrease as the polymer mixes across the boundary layer and where filaments are less pronounced and (iii) the final region where the polymer mixing and dilution is set by the rate of boundary layer growth. Unlike pipe-flow friction-drag reduction, the asymptotic maximum drag reduction (MDR) either was not reached or did not persist in these experiments. Instead, the nearest approach to MDR was transitory and occurred between the development and transitional regions. The length of the development region was observed to increase monotonically with increasing polymer molecular weight, injection rate, concentration and decreasing free stream speed. And finally, the near-wall polymer concentration is correlated to the measured drag reduction for the three polymer molecular weights in the form of a proposed empirical drag-reduction curve.

A hybrid boundary integral/slender body algorithm for modelling flagellar cell motility is presented. The algorithm uses the boundary element method to represent the ‘wedge-shaped’ head of the human sperm cell and a slender body theory representation of the flagellum. The head morphology is specified carefully due to its significant effect on the force and torque balance and hence movement of the free-swimming cell. The technique is used to investigate the mechanisms for the accumulation of human spermatozoa near surfaces. Sperm swimming in an infinite fluid, and near a plane boundary, with prescribed planar and three-dimensional flagellar waveforms are simulated. Both planar and ‘elliptical helicoid’ beating cells are predicted to accumulate at distances of approximately 8.5–22 μm from surfaces, for flagellar beating with angular wavenumber of 3π to 4π. Planar beating cells with wavenumber of approximately 2.4π or greater are predicted to accumulate at a finite distance, while cells with wavenumber of approximately 2π or less are predicted to escape from the surface, likely due to the breakdown of the stable swimming configuration. In the stable swimming trajectory the cell has a small angle of inclination away from the surface, no greater than approximately 0.5°. The trapping effect need not depend on specialized non-planar components of the flagellar beat but rather is a consequence of force and torque balance and the physical effect of the image systems in a no-slip plane boundary. The effect is relatively weak, so that a cell initially one body length from the surface and inclined at an angle of 4°–6° towards the surface will not be trapped but will rather be deflected from the surface. Cells performing rolling motility, where the flagellum sweeps out a ‘conical envelope’, are predicted to align with the surface provided that they approach with sufficiently steep angle. However simulation of cells swimming against a surface in such a configuration is not possible in the present framework. Simulated human sperm cells performing a planar beat with inclination between the beat plane and the plane-of-flattening of the head were not predicted to glide along surfaces, as has been observed in mouse sperm. Instead, cells initially with the head approximately 1.5–3 μm from the surface were predicted to turn away and escape. The simulation model was also used to examine rolling motility due to elliptical helicoid flagellar beating. The head was found to rotate by approximately 240° over one beat cycle and due to the time-varying torques associated with the flagellar beat was found to exhibit ‘looping’ as has been observed in cells swimming against coverslips.

We investigate numerically vortex-induced vibrations (VIV) of two identical two-dimensional elastically mounted cylinders in tandem in the proximity–wake interference regime at Reynolds number Re = 200 for systems having both one (transverse vibrations) and two (transverse and in-line) degrees of freedom (1-DOF and 2-DOF, respectively). For the 1-DOF system the computed results are in good qualitative agreement with available experiments at higher Reynolds numbers. Similar to these experiments our simulations reveal: (1) larger amplitudes of motion and a wider lock-in region for the tandem arrangement when compared with an isolated cylinder; (2) that at low reduced velocities the vibration amplitude of the front cylinder exceeds that of the rear cylinder; and (3) that above a threshold reduced velocity, large-amplitude VIV are excited for the rear cylinder with amplitudes significantly larger than those of the front cylinder. By analysing the simulated flow patterns we identify the VIV excitation mechanisms that lead to such complex responses and elucidate the near-wake vorticity dynamics and vortex-shedding modes excited in each case. We show that at low reduced velocities vortex shedding provides the initial excitation mechanism, which gives rise to a vertical separation between the two cylinders. When this vertical separation exceeds one cylinder diameter, however, a significant portion of the incoming flow is able to pass through the gap between the two cylinders and the gap-flow mechanism starts to dominate the VIV dynamics. The gap flow is able to periodically force either the top or the bottom shear layer of the front cylinder into the gap region, setting off a series of very complex vortex-to-vortex and vortex-to-cylinder interactions, which induces pressure gradients that result in a large oscillatory force in phase with the vortex shedding and lead to the experimentally observed larger vibration amplitudes. When the vortex shedding is the dominant mechanism the front cylinder vibration amplitude is larger than that of the rear cylinder. The reversing of this trend above a threshold reduced velocity is associated with the onset of the gap flow. The important role of the gap flow is further illustrated via a series of simulations for the 2-DOF system. We show that when the gap-flow mechanism is triggered, the 2-DOF system can develop and sustain large VIV amplitudes comparable to those observed in the corresponding (same reduced velocity) 1-DOF system. For sufficiently high reduced velocities, however, the two cylinders in the 2-DOF system approach each other, thus significantly reducing the size of the gap region. In such cases the gap flow is entirely eliminated, and the two cylinders vibrate together as a single body with vibration amplitudes up to 50% lower than the amplitudes of the corresponding 1-DOF in which the gap flow is active. Three-dimensional simulations are also carried out to examine the adequacy of two-dimensional simulations for describing the dynamic response of the tandem system at Re = 200. It is shown that even though the wake transitions to a weakly three-dimensional state when the gap flow is active, the three-dimensional modes are too weak to affect the dynamic response of the system, which is found to be identical to that obtained from the two-dimensional computations.

Plane Couette flow transitions to turbulence at Re ≈ 325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier–Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.