Motivated by observations of Williamson & Roshko of the wake of an oscillating cylinder with three vortices per cycle, and by the analyses of Rott and Aref of the motion of three vortices with vanishing net circulation on the unbounded plane, the integrable problem of three interacting, periodic vortex rows is solved. The problem is ‘mapped’ onto a problem of advection of a passive particle by a certain set of fixed point vortices. The results of this mapped problem are then re-interpreted in terms of the motion of the vortices in the original problem. A rather complicated structure of the solution space emerges with a surprisingly large number of regimes of motion, some of them somewhat counter-intuitive. Representative cases are analysed in detail, and a general procedure is indicated for all cases. We also trace the bifurcations of the solutions with changing linear momentum of the system. For rational ratios of the vortex circulations all motions are periodic. For irrational ratios this is no longer true. The point vortex results are compared to the aforementioned wake experiments and appear to shed light on the experimental observations. Many additional possibilities for the wake dynamics are suggested by the analysis.

The factor which is of prime importance in influencing the shock reflection geometry, and resulting pressures, following impingement of a shock wave on a porous surface is the velocity of the flow into the surface. A set of experiments has been conducted, using holographic inferometry in a shock tube, on the impingement of a shock wave on a surface covered with slits, over the full range of shock incidence angles from 0 to 90°. Inverse shock pressure ratios of 0.4, 0.5 and 0.7 were used, and detailed characterization of the flow fields determined. A number of methods are used to infer the inflow into the surface, and measurements are also conducted on the downstream side of the slit plate in order to establish the pressure ratio across the plate. The tests include choking of the flow through the slits. Shock reflection angles are found to be depressed compared to reflection from an impervious wall for cases of regular reflection, but are similar in the case of Mach reflection with the incident wave near glancing incidence. Contrary to assumptions made in previous work it is shown that for wall angles from zero up to approximately 60° the inflow to the plate is inclined to the surface at about 17° and then tends to straighten out until, for normal shock reflection, the flow is also normal to the plate. It appears that this behaviour is linked to the separation of the flow at the inlet to the pores of the model, due to shock wave diffraction. The maximum value of the absolute inflow velocity occurs in the region of transition from regular to Mach reflection. A series of starting vortices is shed on the underside of the slit and is found to follow a path nearly normal to the plate. These vortices lie along a contact surface whose motion is compatible with the strength of the shock wave transmitted through the plate.

Late wakes (Nt > 20) of towed spheres in a stably stratified fluid were analysed in a plane using a reliable, customized DPIV technique that provides sufficient spatial and temporal resolution to cover all important scales of motion in this freely decaying geophysical flow. Systematic experiments were conducted with independent variation of Re ∈ [103, 104] and F ∈ [1, 10] (F ≡ 2U/ND is an internal Froude number based on the buoyancy frequency, N, and the sphere radius, D/2), and for selected {Re, F} pairs above this range.

The normalized wake width grows at approximately the same rate as in a three-dimensional unstratified wake, but it becomes narrower, not wider, with decreasing F (i.e. as stratification effects become more important). The centreline defect velocity, on the other hand, reaches values an order of magnitude above those measured for three-dimensional unstratified wakes at equivalent downstream locations. Both observations are argued to be consequences of the very high degree of order and coherence that emerge in the late-wake vortex structures.

Streamwise-averaged turbulence quantities, such as the velocity fluctuation magnitude, and mean-square enstrophy, show similar power law behaviour for all Re ≤ 5 × 103, with exponents equal to those expected in three-dimensional axisym-metric turbulent wakes. There is no obvious physical reason why three-dimensional arguments are so successful in such a flow, and at such long evolution times. The scaling collapses none of the cases for Re below 4 – 5 × 103, appearing to establish a minimum Re for a class of self-similar stratified wake flows that evolve from fully turbulent initial conditions.

Individual vortex cross-sections appear to be well approximated by Gaussian distributions at all Re, F and Nt studied here. The scaling behaviour of individual vortices mimics that of the statistical, wake-averaged quantities, and differs measurably from a simple two-dimensional viscous diffusion model. The importance of formulating a realistic three-dimensional model is discussed, and some limited steps in this direction point to future useful experiments and modelling efforts.

Free-convective flow in the neighbourhood of the upper pole of a heated sphere at high Grashof number is considered. Direct buoyancy effects have been studied previously, and it is known that the unsteady boundary-layer solution may terminate in singular behaviour. For a spatially varying surface temperature, a self-induced pressure gradient is present. The effects of this are examined in detail, and it is shown that a singular behaviour may again terminate the solution. The competing effects of buoyancy and induced pressure gradient are examined to delineate those cases in which a steady-state flow is achieved.

A series expansion method is employed to determine the first-order terms in curvature ε and torsion η of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers, De = Re ε1/2, up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germano numbers, Gn = η Re, up to at least 50 where Re is the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in ε and η neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently small Re is dominated by torsion effects; and that for increasing Re, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall. For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.

We show that two-dimensional ‘point’ vortex dynamics in both a polytropic fluid of γ = 3/2 and an isothermal fluid stratified by a constant gravitational field can be written in Hamiltonian form. We find that the formulation admits only one constant of the motion in addition to the Hamiltonian, so that two vortices are the most for which the motion is generally integrable. We study in detail the two-vortex problem and find a rich collection of behaviour: closed trajectories analogous to the circular orbits of the uniform-fluid two-vortex problem, open trajectories for which the self-propelled vortices scatter off each other, and both unstable and stable steadily translating pairs of vortices. Comparison is made to the case of two vortices in a uniform-density fluid bounded by a wall.

A closed-form analytic solution for the motion of axisymmetric rigid pellets suspended in a Newtonian fluid and driven under a pressure gradient through a rigid impermeable cylindrical tube lined with a porous deformable biphasic wall layer is derived using mixture and lubrication theories. The analysis details the velocity distributions in the lubrication and wall layers as well as the solid-phase displacement field in the wall layer. Expressions for the shear stress and pressure gradient are obtained throughout the lubrication and wall layers. Results are presented in terms of resistance, volume flow, and driving pressure relative to smooth-walled tubes for cases both with and without rigid spheres flowing in the free lumen. The analysis is motivated by its possible relevance to the rheology of blood in the microcirculation wherein the endothelial-cell glycocalyx – a carbohydrate-rich coat of macromolecules consisting of proteoglycans and glycoproteins expressed on the luminal surface of the capillary wall – might exhibit similar behaviour to the wall layer modelled here. Estimates of the permeability of the glycocalyx are taken from experimental data for fibrinogen gels formed in vitro. In a tube without pellets lined with a porous wall layer having a thickness which is 15% of the tube radius and having a permeability in the range of fibrinogen gels, approximately a 70% greater pressure drop is required to achieve the same volume flow as would occur in an equivalent smooth-walled tube without a wall layer. If, in the presence of this same wall layer, a rigid spherical pellet is introduced which is 99.5% of the free-lumen radius, the apparent viscosity increases by as much as a factor of four with a concomitant reduction in tube hematocrit of about 10% relative to the corresponding values in an equivalent smooth-walled tube having the same sphere-to-tube diameter ratio without a wall layer.

The collapse of a compressed elastic tube conveying a flow occurs in several physiological applications and has become a problem of considerable interest. Laboratory experiments on a finite length of collapsible tube reveal a rich variety of self-excited oscillations, indicating that the system is a complex, nonlinear dynamical system. Following our previous study on steady flow in a two-dimensional model of the collapsible tube problem (Luo & Pedley 1995), we here investigate the instability of the steady solution, and details of the resulting oscillations when it is unstable, by studying the time-dependent problem. For this purpose, we have developed a time-dependent simulation of the coupled flow – membrane problem, using the Spine method to treat the moving boundary and a second-order time integration scheme with variable time increments.

It is found that the steady solutions become unstable as tension falls below a certain value, say Tu, which decreases as the Reynolds number increases. As a consequence, steady flow gives way to self-excited oscillations, which become increasingly complicated as tension is decreased from Tu. A sequence of bifurcations going through regular oscillations to irregular oscillations is found, showing some interesting dynamic features similar to those observed in experiments. In addition, vorticity waves are found downstream of the elastic section, with associated recirculating eddies which sometimes split into two. These are similar to the vorticity waves found previously for flow past prescribed, time-dependent indentations. It is speculated that the mechanism of the oscillation is crucially dependent on the details of energy dissipation and flow separation at the indentation.

As tension is reduced even further, the membrane is sucked underneath the downstream rigid wall and, although this causes the numerical scheme to break down, it in fact agrees with another experimental observation for flow in thin tubes.

Steady developing flow of an incompressible Newtonian fluid in a curved duct of square cross-section (the Dean problem) is investigated both experimentally and numerically. This study is a continuation of the work by Bara, Nandakumar & Masliyah (1992) and is focused on flow rates between Dn = 200 and Dn = 600 (Dn = Re/(R/a)1/2, where Re is the Reynolds number, R is the radius of curvature of the duct and a is the duct dimension; the curvature ratio, R/a, is 15.1).

Numerical simulations based on the steady three-dimensional Navier – Stokes equations predict the development of a 6-cell secondary flow pattern above a Dean number of 350. The 6-cell state consists of two large Ekman vortices and two pairs of small Dean vortices near the outer wall that result from the primary instability that is of centrifugal nature. The 6-cell flow state develops near θ = 80° and breaks down symmetrically into a 2-cell flow pattern.

The apparatus used to verify the simulations had a duct dimension of 1.27 cm and a streamwise length of 270°. At a Dean number of 453, different velocity profiles of the 6-cell flow state at θ = 90° and spanwise profiles of the streamwise velocity at every 20° were measured using a laser-Doppler anemometer. All measured velocity profiles, as well as flow visualization of secondary flow patterns, are in very good agreement with the simulations, indicating that the parabolized Navier – Stokes equations give an accurate description of the flow.

Based on the similarity with boundary layer flow over a concave wall (the Görtler problem), it is suggested that the transition to the 6-cell flow state is the result of a decreasing spanwise wavelength of the Dean vortices with increasing flow rate. A numerical stability analysis shows that the 6-cell flow state is unconditionally unstable. This is the first time that detailed experiments and simulations of the development of a 6-cell flow state are reported.

Mass transport under a progressive sea wave propagating over a rippled bed is investigated. Wave amplitudes a* of the same order of magnitude as that of the boundary layer thickness δ* and of the ripple wavelength l* are considered. All the above quantities are assumed to be much smaller than the wavelength L* of the sea wave and much larger than the amplitude 2ε* of the ripples. The analysis is carried out up to the second order in the wave slope a*/L* and in the parameter ε*/δ* which is a measure of ripple steepness. Because of these assumptions, the slow damping of wave amplitude in the direction of wave propagation is taken into account. Attention is focused on the bottom boundary layer where an order (ε*/δ*)2 correction of the steady velocity components described by Longuet-Higgins (1953) is found. This correction persists at the outer edge of the bottom boundary layer and affects the solution in the entire water column.

We calculate steady solutions of the Euler equations for any given value of energy and circulation (and angular momentum in the case of a circular domain). A linear relationship between vorticity and stream function is assumed. These solutions correspond to the predicted self-organization into a maximum-entropy state, in the limit of strong mixing. Vorticity mixing is then only weakly restricted by the constraint of energy conservation. While maximum-entropy solutions depend in general on the whole probability distribution of vorticity levels, these linearized results depend only on a single control parameter, yet keeping much of the general structure of the problem. A convenient classification of the maximum-entropy states is thus provided. We show furthermore how to extend these linearized results as an expansion in energy, involving successive moments of the vorticity probability distribution. They are applied to a rectangular domain and compared with existing numerical and laboratory results. We predict that the flow organizes into a single vortex in the square domain, but into a two-vortex dipolar state in a rectangle with aspect ratio greater than 1.12. The case of a circular domain is also explicitly solved, taking into account the conservation of the angular momentum.

This paper presents several features at supercritical conditions of natural convection of large Prandtl number fluids in a vertical slot. Both experimental and numerical results show that even at slightly supercritical conditions the multicellular flow is in an unsteady state which is attributed to the moving and merging of cells. The effect of variable viscosity on the unsteadiness is investigated numerically. Successive merging and appearance of cells is characteristic of this convection so that the number of secondary cells changes with time. The bifurcation phenomena characterized by several modes of unsteady cellular flow are examined.

Two direct numerical simulations of several buoyant bubbles in a two-dimensional periodic domain are presented. The average rise Reynolds number of the bubbles is close to 2, and surface tension is high, resulting in small bubble deformation. The void fraction is relatively high, and the bubbles interact strongly. Simulations of the motion of both 144 and 324 bubbles show the formation of flow structures much larger than the bubble size, and a continuous increase in the energy of the low-wavenumber velocity modes. Plots of the energy spectrum show a range of wavenumbers with an approximately -5/3 slope. This suggests that a part of the work done by the buoyant bubbles is not dissipated, but instead increases the energy of flow structures much larger than the bubbles. This phenomenon, which is also seen in numerical simulation of forced two-dimensional turbulence, prevents the appearance of a statistically steady-state motion that is independent of the size of the computational domain.

This paper presents a theoretical investigation of the occurrence of hydraulic jumps in two-layer systems induced by extraction of fluid from the upper layer. The physical configuration consists of a horizontal main pipe along which air and water flow, and a vertically upward side arm. An hydraulic model based on the momentum principle assuming that the fluids do not mix is developed that leads to at least two possible conjugate states for any given two-layer flow. A method of determining the amount of gas which must be extracted into the side arm for a jump to occur is developed and predictions shown to be in reasonable agreement with observation. Unusually, it is shown that above this critical gas take-off value two possible states remain energetically feasible.

Euromech colloquium 339 was organized by C. Staquet in Lyon (France) from September 6 to 9, 1995. It involved seventy-six participants from fourteen countries. Papers were presented on various aspects of stably stratified flows: (i) internal waves, their generation mechanisms, propagation and reflection properties, their instabilities leading to breaking; (ii) vortex structures in stably stratified fluids, which can be slow layerwise structures, or small intense vortices, appearing for instance in shear flow instabilities; (iii) statistical properties of random wave fields or stratified turbulence; (iv) mixing properties resulting from internal wave fields and stratified turbulence. These quite different dynamical regimes are often closely connected in actual flow problems, and one purpose of this colloquium was to better understand these connections. Participants were interested in fundamental aspects or in more specific applications, in engineering, geophysics and astrophysics. The colloquium was a rare opportunity to gather together scientists with these different points of view, to compare approaches and results, and to highlight general problems.

In this paper, the results of experiments on unsteady disturbances in the boundary-layer flow over a disk rotating in otherwise still air are presented. The flow was perturbed impulsively at a point corresponding to a Reynolds number R below the value at which transition from laminar to turbulent flow is observed. Among the frequencies excited are convectively unstable modes, which form a three-dimensional wave packet that initially convects away from the source. The wave packet consists of two families of travelling convectively unstable waves that propagate together as one packet. These two families are predicted by linear-stability theory: branch-2 modes dominate close to the source but, as the packet moves outwards into regions with higher Reynolds numbers, branch-1 modes grow preferentially and this behaviour was found in the experiment. However, the radial propagation of the trailing edge of the wave packet was observed to tend towards zero as it approaches the critical Reynolds number (about 510) for the onset of radial absolute instability. The wave packet remains convectively unstable in the circumferential direction up to this critical Reynolds number, but it is suggested that the accumulation of energy at a well-defined radius, due to the flow becoming radially absolutely unstable, causes the onset of laminar–turbulent transition. The onset of transition has been consistently observed by previous authors at an average value of 513, with only a small scatter around this value. Here, transition is also observed at about this average value, with and without artificial excitation of the boundary layer. This lack of sensitivity to the exact form of the disturbance environment is characteristic of an absolutely unstable flow, because absolute growth of disturbances can start from either noise or artificial sources to reach the same final state, which is determined by nonlinear effects.