A rotating suspension of non-neutrally buoyant particles, confined by a horizontal cylinder, can be unstable to axial perturbations in concentration. A highly regular pattern of particle density and fluid flow then coexists in a non-equilibrium stationary state. The density profile along the cylinder axis is roughly sinusoidal, with a well-defined wavelength close to the cylinder diameter, and has a magnitude of approximately 30% of the average number density. We have used numerical simulations within the Stokes-flow approximation to investigate the mechanism underlying axial-band formation. Our results show that bands develop from an inhomogeneous particle distribution in the radial plane, which is itself driven by the competition between gravity and the viscous drag of the rotating fluid. We have discovered that the mean angular velocity of the particles is an order parameter which distinguishes between a low-frequency segregated phase and a high-frequency dispersed phase, where the particles fill the whole volume uniformly. The order parameter is a function of a single dimensionless frequency, with a characteristic length that is the mean interparticle separation. As the rotational frequency increases, the particle distribution becomes more homogeneous and the band structure disappears. Hydrodynamic diffusion stabilizes the suspension against centrifugal forces, allowing for a uniformly dispersed phase that can be used to grow three-dimensional cell cultures in an artificial microgravity environment.

The 41st EUROMECH Colloquium on flows with concentrated vorticity was held in Norwich from 17 to 21 September 1973. There were sixty-five participants from nine countries and the author was the chairman of the organizing committee.

From analytical investigations it is well known that the roll-up of an inviscid plane vortex sheet which separates at the edge of a body is a self-similar process which can be described by scaling laws. Unlike plane vortices, ring vortices have a curved rotational axis. For this special vortex type experimental investigations as well as calculations in the literature suggest that the scaling laws are only partially valid. The main goal of this work is to clarify how far these similarity or scaling laws are also valid for the formation of viscid laminar vortex rings. Therefore, the formation process of laminar vortex rings was investigated numerically using a CFD (computational-fluid-dynamics) code. The calculations refer to an experimental setup for which detailed experimental data are available in the literature. In this setup, laminar ring vortices are generated by ejecting water from a circular tube into a quiescent environment by means of a piston. First, a case based on a constant piston velocity was investigated. Comparing calculated and measured data yields a very good agreement. Further calculations were made when forcing the velocity of the piston by three different time-dependent functions. The results of these calculations show that the formation laws for inviscid plane vortices are also valid for the formation process of viscid ring vortices. This applies to the normalized axial and radial position of the vortex centre as well as the normalized diameter of the vortex spiral. However, the similarity laws are valid only if the process is considered in a special frame of reference which moves in conjunction with the front of the jet and if the starting time of the formation process with respect to the starting time of the ejection is taken into account. Additionally, the formation of a ring vortex, which occurs during the start-up process of a free jet flow, was calculated. The results confirm a dependence for the motion of the jet front, which is known from analytical considerations and allows some interesting features to be identified.

This paper describes a series of experiments carried out in a high-enthalpy stream of argon on materials that are known to sublime. The results confirm that an axisymmetric Teflon model ablates to a stable shape which is independent of the initial nose profile. The effect of changing the total enthalpy of the gas is simply to alter the recession rate of the nose. The experimental results show poor agreement with a simple theory which ignores the effects of mass transfer in the boundary layer.

A quantitative analogy between fully developed turbulent flows in curved pipes and orthogonally rotating pipes will be described through similarity arguments, the use of experimental data and computational results. A pair of similarity parameters will be derived for each turbulent flow, so that they have the same dynamical meaning as those of laminar flows. When the second parameter for each flow is large enough, it will be shown that friction factors, as well as heat transfer rates, of the two flows coincide for equal values of the fundamental parameters. Computed contours of velocity and temperature will also reveal strong similarities between the two flows.

We investigate the buoyancy-driven ventilation of an enclosed volume of buoyant fluid, which is connected to the exterior through two openings at the top of the enclosure. An exchange flow becomes established, with outflow through one opening being matched by an equal and opposite inflow through the other vent. The inflowing flux of dense exterior fluid develops a turbulent buoyant plume, which mixes with the interior fluid as it cascades to the base of the fluid volume. An upward return flow gradually stratifies the confined volume of fluid, with a first front of relatively dense plume fluid advancing to the top of the space. Initially, the exchange flow is steady, but as the first front rises above the inflow opening, the flow rate wanes. The initial development of the exchange flow and interior stratification follows the classical filling box work of Baines & Turner (J. Fluid Mech., vol. 37, 1969, p. 51), with a plume of constant buoyancy flux. Once the first front exits the space, the volume flux rapidly asymptotes to the form Q = Q0(τ/(τ+t)) from the initial value Q0, where τ is the buoyancy-driven draining time, based on the initial density contrast with the environment. The vertical structure of the interior density stratification asymptotes to a profile of constant shape whose amplitude decays in time as (τ/(τ+t))2, and we identify conditions under which the vertical variation in density is comparable to the difference between mean interior density and the exterior. This generalises the analysis presented by Linden, Lane-Serff & Smeed (J. Fluid Mech., vol. 212, 1990, p. 309), who assumed that the interior fluid is well mixed. New laboratory experiments of the process are shown to be consistent with our predictions of the evolution of the flow, the interior stratification and the migration of contaminants. We also develop our analysis for situations where there are multiple stacks and show how this improves the mixing efficiency for a given ventilation flow. The model has relevance for the design of transient mixing ventilation in a building, especially when the effect of vertical stratification is important for ensuring thermal comfort.

This study deals with the mass-transport velocity within the bottom boundary layer of cnoidal waves progressing over a smooth horizontal bed. Mass-transport velocity distributions through the boundary layer are derived and compared with that predicted by Longuet-Higgins (1953) for sinusoidal waves. The mass transport at the outer edge of the boundary layer is compared with various theoretical results for an inviscid fluid based on cnoidal wave theory and also with previous experimental results. The effect of the viscous boundary layer is to establish uniquely the bottom mass transport and this is appreciably greater than the somewhat arbitrary prediction for an inviscid fluid.

The general solutions of the Stokes approximate equations of motion are derived for two-dimensional and axisymmetric flows in the half-space x > 0, for an arbitrarily given velocity field in the plane x = 0. There is assumed to be no solid surface in the half-space. According to whether the velocity at infinity is zero or not, the solutions can be said to describe either jet-type or wake-type flows. Only the latter category is considered; numerical examples are worked out and properties of the base flow at very low Reynolds numbers are investigated. A recirculating flow region may exist, but the flow properties are not sensitive to this feature.

This study proposes an onset condition of shock-free supersonic vortex breakdown from the axial momentum variation, which applies in the presence or absence of a stagnation point. The condition is derived from a comprehensive approach to vortex breakdown. Supersonic breakdown appeared when the swirl parameter and Mach number were small. Moreover, bubble-type breakdowns with a stagnation point, which occur in subsonic conditions, could not occur under the supersonic condition in the present analysis. The predicted breakdowns under this condition were consistent with the results of the three-dimensional numerical simulations for Mach numbers ranging from 1.5 to 5.0. Supersonic vortex breakdowns were clearly captured by the helicity contours in the numerical results. The threshold of the downstream Mach number required for spiral breakdown with no stagnation point was also theoretically derived and verified in numerical results. These findings provide new insights into vortex breakdown in supersonic flows.

We present direct numerical simulations of the spatial development of normal mode perturbations to boundary layers with Falkner–Skan velocity profiles. Values of the pressure gradient parameter considered range from very small, i.e. nearly flat-plate conditions, to relatively large values corresponding to incipient separation. In almost all cases, we find that the most effective perturbation is one composed of a plane wave and a pair of oblique waves inclined at equal and opposite angles to the primary flow direction. The frequency of the oblique waves is half that of the fundamental plane wave and because the conditions for resonance are satisfied exactly, all modes share a common critical layer, thus facilitating a strong interaction.

The oblique waves initially undergo a parametric type of subharmonic resonance, but in accordance with recent analyses of non-equilibrium critical layers, the system subsequently becomes fully coupled. From that point on, the amplification of all modes, including the plane wave, substantially exceeds the predictions of linear stability theory. Good agreement is obtained with the experimental small pressure gradient results of Corke & Gruber (1996). Our growth rates are slightly larger flowing to slight differences in initial conditions (e.g. the angle of inclination of the oblique waves).

The spectral element method was used to discretize the Navier–Stokes equations and the preconditioned conjugate gradient method was used to solve the resulting system of algebraic equations. At the inflow boundary, Orr–Sommerfeld modes were employed to provide the initial forcing, whereas the buffer domain technique was used at the outflow boundary to prevent convective wave reflection or upstream propagation of spurious information.

In many tidal embayments, complex patterns of channels and shoals are observed. To gain a better understanding of these features, an idealized model, that describes the interaction of water motion, sediment transport and bed evolution in a semi-enclosed, rectangular basin, is developed and analysed. To explain the initial formation of channels and shoals, two-dimensional perturbations superposed on a laterally uniform equilibrium bottom are studied. These perturbations evolve due to convergences of various residual suspended sediment fluxes: a diffusive flux, a flux related to the bed topography, an advective flux resulting from internally generated overtides and an advective flux due to externally prescribed overtides. For most combinations of these fluxes, perturbations start to grow if the bottom friction is strong enough. Their growth is mainly a result of convergences of diffusive and topographically induced sediment fluxes. Advective contributions due to internally generated overtides enhance this growth. If only diffusive sediment fluxes are considered, the underlying equilibrium is always unstable. This can be traced back to the depth dependence of the deposition parameter. Contrary to the results of previous idealized models, the channels and shoals always initiate in the shallow, landward areas. This is explained by the enhanced generation (compared to that in previous models) of frictional torques in shallow regions. The resulting initial channel–shoal formation compares well with results found in complex numerical model studies. The instability mechanism and the location of the initial formation of bottom patterns do not change qualitatively when varying parameters. Changes are mainly related to differences in the underlying equilibrium profile due to parameter variations.

The inviscid flow behind a cylindrical blast wave and its analogy with hypersonic flow past blunt-nosed cylinders is considered. Sakurai (1953, 1954) obtained a solution for the flow field behind a propagating blast wave by expanding the flow variables in power series of 1/M2, where M is the blast wave Mach number, and determining the coefficients of the first two terms in the series. Here the work is extended to include third-order terms. Third-order theory is shown to improve the prediction of shock wave shapes and surface pressure distribution on hemisphere-cylinder configurations at M∞ = 7·7 and 17·18.

A semipermeable membrane forms part of a plane boundary which separates pure solvent from a solution of concentration Cb. Stirring motions in the solution cause it to flow parallel to the plane with uniform wall shear rate α. The velocity of osmotic solvent flow across the membrane into the solution at any point is PΔC, where P is the osmotic permeability of the membrane and ΔC is the local concentration difference across it. ΔC is reduced below Cb by a position-dependent factor γ because of the concentration boundary layer over the membrane, which is thicker than in the absence of osmosis as a result of advection by the osmotic flow itself. The concentration boundary layer is analysed, on the assumption that it is two-dimensional, for both small and large values of the dimensionless longitudinal co-ordinate \[ \xi = PC_b (9x/\alpha D^2)^{\frac{1}{3}}, \] where x is the dimensional co-ordinate and D is the solute diffusivity. These expansions are used to compute γι, the average value of the flux-reduction factor γ over the whole membrane, as a function of β′, which is the value of ξ at the downstream end of the layer, x = l. It is shown that the standard physiological model, in which the layer has a given thickness δ and the stirring motions are not explicitly considered, gives accurate results for γl as a function of β = PCb δ/D as long as δ is given by \[ \delta = 1.22(Dl/\alpha)^{\frac{1}{3}}, \] so that β is proportional to β′.

A new exact solution of the nonlinear shallow-water equations is presented. The solution corresponds to divergent and non-divergent free oscillations in an infinite straight channel of parabolic cross-section on the rotating Earth. It provides a description of the one-dimensional subclass of shallow-water flows in paraboloidal basins considered by Ball (1964), Thacker (1981), Cushman-Roisin (1987) and others in which the velocity field varies linearly and the free-surface displacement varies quadratically with the spatial coordinates. In contrast to the previous exact solutions describing divergent oscillations in circular and elliptic paraboloidal basins, the oscillation frequency of the divergent oscillation in the parabolic channel is found to depend, in part, on the amplitudes of the relative vorticity and free-surface curvature. This result is consistent with Thacker's (1981) numerical finding that when the free surface in parabolic channel flow is curved, the oscillation frequency depends on the amplitude of the motion. Solutions for parcel trajectories are also presented. The exact solution provides a rare description of a class of nonlinear flows and is potentially valuable as a validation test for numerical shallow-water models in Eulerian and Lagrangian frameworks.

A multiple-timescale analysis is employed to analyse Taylor-dispersion-like convective-diffusive processes in converging and diverging flows. A long-time asymptotic equation governing the cross-sectionally averaged solute probability density is derived. The form of this equation is shown to be dependent upon the number of spatial dimensions characterizing the duct or ‘cone’. The two-dimensional case (non-parallel plates) is shown to be fundamentally different from that for three dimensions (circular cone) in that, in two dimensions, a Taylor dispersion description of the process is possible only for small Peclet numbers or angles of divergence. In contrast, in three dimensions, a Taylor dispersion description is always possible provided sufficient time has passed since the initial introduction of solute into the system. The convective Taylor dispersion coefficients $\overline{D}_c$ for the respective cases of low-Reynolds-number flow between non-parallel plates and in a circular cone are computed and their limiting values, $\overline{D}_c^0$, for zero apex angle are shown to be consistent with the known results for Taylor dispersion between parallel plates and in a circular cylinder. When plotted in the non-dimensional form of $\overline{D}_c/\overline{D}^0_c$ versus the half-vertex angle θ0, the respective dispersivity results for the two cases hardly differ from one another, increasing monotonically from 1.0 for θ0 = 0 to approximately 2.6 for a fully flared duct, θ0 = θ/2. Lastly, the techniques developed above for the case of rectilinear channel and duct boundaries are extended to the case of curvilinear boundaries, and an illustrative calculation performed for the case of axisymmetric flow in a flared Venturi tube.

Irrotational flows caused by a body moving with a constant velocity in an unbounded homentropic compressible fluid at rest at infinity are considered. Provided the (steady) flow relative to the body is everywhere subsonic, it is shown that the momentum mass is always equal to the drift mass, and the kinetic-energy mass is equal to the drift mass under certain conditions.

We consider the inviscid stability of the Batchelor (1964) vortex in a compressible flow. The problem is tackled numerically and also asymptotically, in the limit of large (azimuthal and streamwise) wavenumbers, together with large Mach numbers. The nature of the solution passes through different regimes as the Mach number increases, relative to the wavenumbers. At very high wavenumbers and Mach numbers, the mode which is present in the incompressible case ceases to be unstable, whilst a new ‘centre mode’ forms, whose stability characteristics are determined primarily by conditions close to the vortex axis. We find that generally the flow becomes less unstable as the Mach number increases, and that the regime of instability appears generally confined to disturbances in a direction counter to the direction of the rotation of the swirl of the vortex.

Throughout the paper comparison is made between our numerical results and results obtained from the various asymptotic theories.