Some insight into the behaviour of tightly fitting solid pellets, which may be deformable, and are being forced by a pressure difference to move slowly along a distensible tube filled with viscous fluid, is sought by theoretical study of a simple axisymmetric model (§2). In this, the pellet's clearance in the tube is taken to be a small fraction of the tube radius; the fraction may, at a pressure characteristic of that ahead of the pellet, be either positive or negative. Even if it is positive, the tube may still be distended (or the pellet compressed, or both) as the pellet passes, because the thickness of lubricating film generated may exceed the clearance. Naturally, still greater elastic deformation can occur in the case of negative clearance.

Highly simplified elastic properties are assumed; with an eye on tubes occurring in physiological systems (with Poisson's ratio close to 0.5), the local distension of the tube is taken to vary linearly with the local excess pressure; as a still cruder approximation, a similar relation for local reduction of pellet radius is assumed. A parabolic approximation to the pellet's undistorted meridian section, in the region where the lubricating film is thin, is also assumed, leading to a simple relation between pressure and local film thickness which is used, together with Reynolds's lubrication equation, to evaluate both. An arbitrary constant, the rate of leakback of fluid past the pellet, is determined by the condition that the pressure difference forcing the pellet must just balance the skin-frictional resistance to its motion.

The problem is non-dimensionalized (§3) and reduced to that of finding a particular solution of a differential equation containing a certain parameter L. In addition to numerical solutions for particular values of L (§6), perturbation solutions for both small and large L are obtained (§§4 and 5), to give mathematical and physical insight; the perturbation for large L (corresponding to negative clearance) is highly singular, requiring the matching of approximate solutions different in each of six different layers.

A striking feature of the solutions is a necking of the gap between pellet and tube behind the pellet. This is so pronounced in the case of negative clearance (figure 2) that it might give the false impression that the pellet was being propelled by peristaltic contraction of the tube instead of by fluid pressure gradient. The physical reason for this is elucidated (§6).

In the case of positive clearance, rather small pressure differences suffice, on this theory, to propel the pellet, because different parts of the lubricating layer act on it with frictional resistances of different signs, which almost cancel out. By contrast, for negative clearance, the resistance becomes a large multiple of that found in a purely fluid-filled tube of length and mean velocity equal to that of the pellet. This multiple increases, and the film thickness correspondingly decreases (figure 7), as the pellet velocity decreases.

One physiological system on which the model may throw some light is the narrow capillary with red blood cells being squeezed through it in single file, lubricated by plasma (§1 and 8). At the higher flow speeds, around 0.1 mm/s, the lubricating film, predicted to be about 0·2μm thick, appears likely to play a significant role in mass transfer to and from the tissue spaces. At much lower speeds, predicted film thicknesses are so small that any of a number of mechanisms, including loss of fluid through the porous capillary wall due to the local excess pressure in the layer, might cause movement to ‘seize up’ altogether.

Measurements of grid turbulence have been obtained for grid Reynolds numbers ranging from 2·4 × 106 to 1·2 × 105. The decay law and the effect of Reynolds number on the turbulence level are established. The measured power spectra of the turbulence are consistent with Kolmogoroff scaling for kη > 0·1 (k is the wave-number and η is the Kolmogoroff length); but for kη < 0·1, the spectra of the stream-wise turbulence velocity component and of the cross-stream component do not appear to be isotropically related. However, the stream-wise spectrum does display a $-\frac{5}{3}$ region, which increases in extent with increasing Reynolds number.

The spatial structure of the streamfunction field of free, linear internal waves in a two-dimensional basin is governed by the canonical, second-order, hyperbolic equation on a closed domain. Its solution can be determined explicitly for some simple shapes of the basin. It consists of an algorithm by which ‘webs’ of uniquely related characteristics can be constructed and the prescription of one (independent) value of a field variable, related to the streamfunction, on each of these webs. The geometric construction of the webs can be viewed as an alternative version of a billiard game in which the angle of reflection equals that of incidence with respect to the vertical (rather than to the normal). Typically, internal waves are observed to be globally attracted (‘focused’) to a limiting set of characteristics. This attracting set can be classified by the number of reflections it has with the surface (its period in the terminology of dynamical systems). This period of the attractor is a fractal function of the normalized period of the internal waves: large regions of smooth, low-period attractors are seeded with regions with high-period attractors. Occasionally, all internal wave rays fold exactly back upon themselves, a ‘resonance’: focusing is absent and a smooth pattern, familiar from the cellular pattern in a rectangular domain, is obtained. These correspond to the well-known seiching modes of a basin. An analytic set of seiching modes has also been found for a semi-elliptic basin. A necessary condition for seiching to occur is formulated.

A study is described of the forced inertial oscillations appearing in an axially rotating completely filled circular cylinder with plane ends. Excitation is provided by causing the top end to rotate about an axis inclined slightly to the rotation axis. Experiments demonstrate the presence of numerous low mode resonances in a densely spaced range of ratios of net cylinder height to radius in close conformance with linear inviscid theory. Where geometry permits simple corner reflexion, characteristic surfaces are revealed which confirm in part the theoretical predictions concerning their scale and form.

Detailed measurements are given of the amplitude at one point within the cylinder for the condition in which the disturbance frequency equals the rotation frequency. Amplitude column height spectra are compared with theoretical estimates, and the evolution of amplitude for the simplest mode of resonant oscillation is studied. A non-linear theory based on the integral energy of large amplitude oscillation is derived whose broad features are in fair quantitative and qualitative agreement with these observations.

Some investigation is made of the phenomenon of resonant collapse, in which larger amplitude resonant oscillations, after persisting in an apparently laminar form, degenerate abruptly into a state of agitation and disorder from which they do not recover. It is found that the time for emergence of this collapse after the introduction of the forcing disturbance has a close correspondence with the theoretical period of one ‘evolutionary’ cycle of momentum exchange between the main motion and the secondary oscillation.

Earlier measurements of the contribution of four distinct classes of motions, i.e. (u < 0, v > 0), (u > 0, v < 0), (u < 0, v < 0) and (u > 0, v > 0), to the Reynolds stress $-\rho \overline{uv}$ in the wall region of a bounded turbulent shear flow have been extended. These classes were obtained by truncating the u and v signals about zero. Various statistical properties of the truncated streamwise and normal velocity components u and v and of their product uv have been determined in an attempt to characterize quantitatively the motions in this flow. Average values and probability density distributions both of the truncated and untruncated signals have been taken.

A new method, herein referred to as optimal mode decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimizes the system residual error. The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized but rather fixed to be the proper orthogonal decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the ‘optimal’ low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.

In stably stratified flow over a three-dimensional hill, we can define a dividing streamline that separates those streamlines that pass around the hill from those that pass over the hill. The height Hs of this dividing streamline can be estimated by Sheppard's simple energy argument; fluid parcels originating far upstream of a hill at an elevation above Hs have sufficient kinetic energy to rise over the top, whereas those below Hs must pass around the sides. This prediction provides the basis for analysing an extensive range of laboratory observations and measurements of stably stratified flow over a variety of shapes and orientations of hills and with different upwind density and velocity profiles. For symmetric hills and small upwind shear, Sheppard's expression provides a good estimate for Hs. For highly asymmetric flow and/or in the presence of strong upwind shear, the expression provides a lower limit for Hs. As the hills become more nearly two-dimensional, these experiments become less well defined because steady-state conditions take progressively longer to be established. The results of new studies are presented here of the development of the unsteady flow upwind of two-dimensional hills in a finite-length towing tank. These measurements suggest that a very long tank would be required for steady-state conditions to be established upstream of long ridges with or without small gaps and cast doubt upon the validity of previous laboratory studies.

Some experimental results on incompressible viscous fluid flow in the gap between two concentric rotating spheres are discussed. The flow field in the spherical gap has been studied qualitatively by flow visualization (photographs) and quantitatively by measurements by the hot-wire technique. For a wide range of Reynolds numbers, the friction torque was measured for several gap widths and a relatively simple method of determining the torque theoretically is given. At higher Reynolds numbers instabilities appear. Their different behaviour for relatively small and large gap widths is demonstrated. For the larger gap widths, the different appearance of the Taylor–Görtler vortices, the reason for their generation, their regimes of existence as well as their influence on the friction torque are thoroughly treated. Detailed information is given on the new effect of the dependence of the wavelength of the vortices on the Reynolds number.

We use a numerical method that was described in Part 1 (Marcus 1984a) to solve the time-dependent Navier-Stokes equation and boundary conditions that govern Taylor-Couette flow. We compute several stable axisymmetric Taylor-vortex equilibria and several stable non-axisymmetric wavy-vortex flows that correspond to one travelling wave. For each flow we compute the energy, angular momentum, torque, wave speed, energy dissipation rate, enstrophy, and energy and enstrophy spectra. We also plot several 2-dimensional projections of the velocity field. Using the results of the numerical calculations, we conjecture that the travelling waves are a secondary instability caused by the strong radial motion in the outflow boundaries of the Taylor vortices and are not shear instabilities associated with inflection points of the azimuthal flow. We demonstrate numerically that, at the critical Reynolds number where Taylor-vortex flow becomes unstable to wavy-vortex flow, the speed of the travelling wave is equal to the azimuthal angular velocity of the fluid at the centre of the Taylor vortices. For Reynolds numbers larger than the critical value, the travelling waves have their maximum amplitude at the comoving surface, where the comoving surface is defined to be the surface of fluid that has the same azimuthal velocity as the velocity of the travelling wave. We propose a model that explains the numerically discovered fact that both Taylor-vortex flow and the one-travelling-wave flow have exponential energy spectra such that In [E(k)] ∝ k1, where k is the Fourier harmonic number in the axial direction.

Turbulence measurements in dispersed-bubble two-phase pipe flow, using laser velocimetry techniques, are presented. The turbulence-intensity measurements show a strong dependence on the quality of the flow. A theoretical basis for the prediction of turbulence levels in two-phase flows is proposed. The approach is applied to dispersed-gas/liquid (bubbly) and solid/gas (particulate) two-phase flows, for which experimental data are available, with excellent results.

Turbulence and secondary flow measurements were undertaken using a two-component laser-Doppler anemometer in meander channels with straight flood plain banks. The most interesting feature of the compound meandering channel flow was found to be the behaviour of the secondary flow. The difference in direction of rotation of the flow before and after inundation at a bend section was confirmed by the detailed velocity measurements. In addition, by performing the measurement over a half wavelength of meander, the originating and developing processes of the secondary flow were also clarified. In contrast to the centrifugal force for inbank flow, the interaction between the main channel flow and the flood plain flow in the cross-over region was found to play an important role in developing a shear produced secondary flow in the overbank cases. New experimental evidence concerning the spatial distribution of Reynolds stress −ρuw, −ρuv and −ρvw are presented for sinuous compound meander channels. In such channels, large interfacial shear stresses were induced at around the bankfull level, especially in the cross-over region, and were found to be larger than the bed shear stress in magnitude. Particular importance is placed on −ρvw, which is usually small compared with other stress components, as the cause of the secondary flow in the lower layer. The influence of secondary flow on eddy viscosity was found also to be significant. These turbulence data are particularly useful in understanding the flow mechanisms that occur in meandering channels and in developing proper turbulence models for such flows.

Electrothermal motion in an aqueous solution arises from the action of an electric field on inhomogeneities in the liquid induced by temperature gradients. The temperature field can be produced by the applied electric field through Joule heating, or caused by external sources, such as strong illumination. Electrothermal flows in microsystems are usually observed at applied signal frequencies around 1 MHz and voltages around 10 V. In this work, we present self-similar solutions for the motion of an aqueous solution in a constant temperature gradient placed on top of: (a) two coplanar electrodes subjected to an a.c. potential difference, and (b) four coplanar electrodes subjected to a four-phase a.c. signal, generating a rotating field. The first case produces two-dimensional rolls whereas the second case produces a liquid whirl. Finally, we present experimental results of electrothermal liquid flows generated by alternating and rotating electric fields under strong illumination, and these experiments are compared to the analytical solutions. The induced rotating flow could be used in the mixing of analytes and of liquids in microsystems.

Measurements, including one-point double, triple or quadruple mean products of velocity fluctuations, have been made in low-speed turbulent, boundary layers on flat surfaces downstream of concave or convex bends with turning angles of 20 or 30 degrees, the length of the curved region being at most 6 times the boundary-layer thickness at entry. These short bends approximate to ‘impulses’ of curvature, and the object of the work was to investigate the impulse response of the boundary layer, essentially the decay of structural changes downstream of the bends. The work can be regarded as a sequel, with much more detailed measurements, to the study by So & Mellor (1972, 1973, 1975) who investigated the response to step increases of curvature: turbulent boundary layers being nonlinear systems, responses to several kinds of curvature history are needed to assemble an adequate description of the flow. The most striking feature of the ‘impulse’ response is that the decay of the high turbulent intensity found at exit from the concave bends is not monotonic; the Reynolds stresses in the outer layer collapse to well below the level at entry, and are still falling slowly at the end of the test rig although in principle they must recover eventually. On the convex (stabilized) side the flow recovers, monotonically in the main, from a low level of turbulent intensity at the exit. The pronounced second-order response on the concave side can be explained qualitatively by interaction between the shear stress and the mean shear and is not peculiar to curved flows, but in the present cases the response is complicated by large changes in the dimensionless structure parameters related to double or triple mean products of velocity fluctuations. Strong spanwise variations, due presumably to longitudinal vortices, further complicate the flow in the concave bends, and decay only very slowly downstream.

This paper describes an experimental study of turbulent boundary layers over two-dimensional spanwise groove and three-dimensional sandgrain roughnesses in the ‘transition regime’ between hydraulically smooth and fully rough conditions. Mean-flow measurements show that a state of kinematic near-self-preservation is also reached by sandgrain roughness and not just by d-type grooved roughness alone as commonly believed; sandgrain roughness simply requires an order-of-magnitude-longer length to reach such a state. The two roughness Reynolds numbers demarcating the boundaries of the transition regime of k-type roughnesses are found to decrease with increasing roughness-element spanwise aspect ratio (span/height). A more important role of the upper-Reynolds-number limit of the transition regime in the drag behaviour is indicated. The two Reynolds-number limits of the transition regime correlate with the two critical Reynolds numbers that describe the stability of the vortex-shedding process existing behind a similar but isolated roughness element lying submerged in an otherwise laminar boundary layer. The results provide a guideline for reducing k-type rough-wall drag by lowering the spanwise aspect ratio of the roughness elements. The vortex-shedding process in rough-wall turbulent boundary layers is described by the stability parameter $U\tau (\overline{T}/\nu)^{\frac{1}{2}}$ whose value is the same for all roughnesses examined herein; here Uτ is the friction velocity, $\overline{T}$ is the mean time period of vortex shedding and v is the kinematic viscosity of the fluid.

Two-dimensional buoyant clouds moving along inclined boundaries under a gravitational force are investigated theoretically and experimentally. It is found that the ‘thermal theory’ gives a good description of the flow in the slope angle range 5° [lsim ] θ ≤ 90°. In this range the spatial growth rates of the cloud height and length are constant for a given slope angle and show a linear dependence on θ. For a cloud released with zero initial velocity the front velocity Uf first increases and then decreases, with the characteristic time of acceleration predicted by theory. In the decelerating state Uf/(g0′Q0/xf)½ is 2·6 ± 0·2 at θ ≃ 15°, and then reduces uniformly with increasing θ to a value of 1·5 ≃ 0·2 at 90° (where g0′Q0 is the released buoyancy and xf is the front position measured from a virtual origin). The shape of the cloud is well approximated by a half-ellipse. The variation of the ratio of the principal axes of the half-ellipse with slope angle is identical with that of the head of an inclined starting plume (Britter & Linden 1980). However, the cloud has a greater growth rate than the head of a starting plume.

We generalize the POD-based Galerkin method for post-transient flow data by incorporating Navier–Stokes equation constraints. In this method, the derived Galerkin expansion minimizes the residual like POD, but with the power balance equation for the resolved turbulent kinetic energy as an additional optimization constraint. Thus, the projection of the Navier–Stokes equation on to the expansion modes yields a Galerkin system that respects the power balance on the attractor. The resulting dynamical system requires no stabilizing eddy-viscosity term – contrary to other POD models of high-Reynolds-number flows. The proposed Galerkin method is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer. Generalizations for more Navier–Stokes constraints, e.g. Reynolds equations, can be achieved in straightforward variation of the presented results.

We present the design and operation of a flow apparatus for investigations of mixing in time-periodic and spatially periodic chaotic flows. Uses are illustrated in terms of two devices operating in the Stokes regime: the partitioned-pipe mixer, a spatially periodic system consisting of sequences of flows in semicircular ducts, and the eccentric helical annular mixer, a time-periodic velocity field between eccentric cylinders with a superposed Poiseuille flow; other mixing flows can be implemented with relative ease. Fundamental differences between spatially periodic and time-periodic duct flows are readily apparent. Steady spatially periodic systems show segregated KAM-tubes coexisting with chaotic advection; such tubes are remarkably stable under a variety of experimental conditions. Time-periodic duct flows lead to complex streakline structures; since regular regions in the cross-sectional flow move through space, a streakline can find itself injected in a regular domain for some time then be trapped in a chaotic region, and so on, leading to ‘intermittent’ behaviour.

The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics. This requires a preliminary description, in terms of q, of the fluid motion beneath the free surface, which kinematical boundary-value problem is solved through a variational formulation and the truncation and inversion of an infinite matrix. The results are applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincaré's action—angle formulation. The general formulation provides for excitation through either horizontal or vertical translation of the basin and for dissipation. Detailed results are given for free and forced oscillations of two, resonantly coupled degrees of freedom.

The dynamics of vorticity in two-dimensional turbulence is studied by means of semi-direct numerical simulations, in parallel with passive-scalar dynamics. It is shown that a passive scalar forced and dissipated in the same conditions as vorticity, has a quite different behaviour. The passive scalar obeys the similarity theory à la Kolmogorov, while the enstrophy spectrum is much steeper, owing to a hierarchy of strong coherent vortices. The condensation of vorticity into such vortices depends critically both on the existence of an energy invariant (intimately related to the feedback of vorticity transport on velocity, absent in passive-scalar dynamics, and neglected in the Kolmogorov theory of the enstrophy inertial range); and on the localness of flow dynamics in physical space (again not considered by the Kolmogorov theory, and not accessible to closure model simulations). When space localness is artificially destroyed, the enstrophy spectrum again obeys a k−1 law like a passive scalar. In the wavenumber range accessible to our experiments, two-dimensional turbulence can be described as a hierarchy of strong coherent vortices superimposed on a weak vorticity continuum which behaves like a passive scalar.

The asymptotic properties of Burgers turbulence at extremely large Reynolds numbers and times are investigated by analysing the exact solution of the Burgers equation, which takes the form of a series of triangular shocks in this situation. The initial probability distribution for the velocity u is assumed to decrease exponentially as u → ∞. The probability distribution functions for the strength and the advance velocity of shocks and the distance between two shocks are obtained and the velocity correlation and the energy spectrum function are derived from these distribution functions. It is proved that the asymptotic properties of turbulence change qualitatively according as the value of the integral scale of the velocity correlation function J, which is invariant in time, is zero, finite or infinite. The turbulent energy per unit length is shown to decay in time t as t−1 (with possible logarithmic corrections) or $t^{-\frac{2}{3}}$ according as J = 0 or J ≠ 0.