The pair distribution function of 3.18 mm diameter particles was measured in the plane of shear of a simple shear flow at concentrations of 5%, 10% and 15% by volume. A new direct flow-visualization procedure and a new pattern recognition algorithm were used in the investigation. The measurements show a depletion of bound pairs of particles in the direction of flow. A simple model which includes the effect of particle surface roughness on the particle interactions and the pair distribution function is presented. An important effect of surface roughness is that the particles in a suspension can experience irreversible interactions in the presence of an externally imposed simple shear flow. The model shows that such irreversibilities eliminate all bound pairs of particles in the plane of shear by displacing particles out of the closed orbit trajectory region. Surface roughness is found to induce significant asymmetry in the fore and aft region of a two-particle interaction. The measurements and predictions are in qualitative agreement with these conclusions.

This paper reports the results of a numerical investigation of the transient evolution of the flow around a spherical bubble rising in a liquid contaminated by a weakly soluble surfactant. For that purpose the full Navier–Stokes equations are solved together with the bulk and interfacial surfactant concentration equations, using values of the physical-chemical constants of a typical surfactant characterized by a simple surface kinetics. The whole system is strongly coupled by nonlinear boundary conditions linking the diffusion flux and the interfacial shear stress to the interfacial surfactant concentration and its gradient. The influence of surfactant characteristics is studied by varying arbitrarily some physical-chemical parameters. In all cases, starting from the flow around a clean bubble, the results describe the temporal evolution of the relevant scalar and dynamic interfacial quantities as well as the changes in the flow structure and the increase of the drag coefficient. Since surface diffusion is extremely weak compared to advection, part of the bubble (and in certain cases all the interface) tends to become stagnant. This results in a dramatic increase of the drag which in several cases reaches the value corresponding to a rigid sphere. The present results confirm the validity of the well-known stagnant-cap model for describing the flow around a bubble contaminated by slightly soluble surfactants. They also show that a simple relation between the cap angle and the bulk concentration cannot generally be obtained because diffusion from the bulk plays a significant role.

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where travelling wave normal modes are assumed as solutions, this work offers a means whereby arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behaviour, including both early-time transients and the long-time asymptotics, can be determined for any initial small-amplitude three-dimensional disturbance. The bases for the theoretical analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. Also, optimal initial conditions measured in terms of the normalized energy growth can be determined in a straightforward manner and such optimal conditions clearly reflect transient growth data that are easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number, it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes when used without regard to a particular initial-value problem. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists (e.g. the Blasius boundary layer). Finally, results from the temporal theory are compared with the equivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spatially accurate direct numerical simulation code.

The extrusion of a yield stress fluid from the space between two parallel plates is investigated experimentally. Oscillating the magnitude of the squeezing force about a mean value (F=f[1+αcos(ωt)]) was observed to significantly enhance the flow rate of yield stress fluids, while having no effect on the flow rate of Newtonian fluids. This is a novel result. The enhancement depends on the magnitude of the force, the oscillatory frequency and amplitude, the fluid being squeezed, and the thickness of the fluid layer. Non-dimensional results for the various flow quantities have been presented by using the flow predicted for the constant-force squeezing of a Herschel–Bulkley yield stress fluid as the reference. In the limit of constant-force squeezing, the present experimental results compare very well with those of our earlier theoretical model for this situation (Zwick, Ayyaswamy & Cohen 1996). The results presented in this paper have significance, among many applications, for injection moulding, in the adhesive bonding of microelectronic chips, and in surgical procedures employed in health care.

The nonlinear stability of a pulsating detonation wave driven by a three-step chain-branching reaction is studied. The reaction model consists sequentially of a chain-initiation step and a chain-branching step, both governed by Arrhenius kinetics, followed by a temperature-independent chain-termination step. The model mimics the essential dynamics of a real chain-branching chemical system, but is sufficiently idealized that a theoretical analysis of the instability is possible. We introduce as a bifurcation parameter the chain-branching cross-over temperature (TB), which is the temperature at which the chain-branching and chain-termination rates are equal. In the steady detonation structure, this parameter controls the ratio of the chain-branching induction length to the length of the recombination zone. When TB is at the lower end of the range studied, the steady detonation structure, which is dominated by the temperature-independent recombination zone, is found to be stable. Increasing TB increases the length of the chain-branching induction region relative to the length of the recombination zone, and a critical value of TB is reached where the detonation becomes unstable, with the detonation shock pressure evolving as a single-mode low-frequency pulsating oscillation. This single-mode nonlinear oscillation becomes progressively less stable as TB is increased further, persisting as the long-term dynamical behaviour for a significant range of TB before eventually undergoing a period-doubling bifurcation to a two-mode oscillation. Further increases in TB lead to a chaotic behaviour, where the detonation shock pressure history consists of a sequence of substantive discontinuous jumps, followed by lower-amplitude continuous oscillations. Finally, for further increases in TB a detonability limit is reached, where during the early onset of the detonation instability, the detonation shock temperature drops below the chain-branching cross-over temperature causing the wave to quench.

Experiments and numerical simulations of fully developed turbulence reveal the existence of elongated vortices whose length is of the order of the integral scale of turbulence while the diameter is somewhere between the Kolmogorov scale and the Taylor microscale. These vortices are embedded in quasi-irrotational background flow whose straining action counteracts viscous decay and determines their cross-sectional shape. In the present paper we analyse the effect of a stretched vortex of this kind on a uni-directional magnetic flux tube aligned with vorticity in an electrically conducting fluid. When the magnetic Prandtl number is large, Pm[gsim ]1, the field is concentrated in a flux tube which, like the vortex itself, has elliptical cross-section inclined at 45° to the principal axes of strain. We focus on the limit Pm[Lt ]1 when the magnetic flux tube has radial extent much larger than that of the vortex, which appears like a point vortex as regards its action on the flux tube. We find the steady-state solution valid in the entire plane outside the vortex core. The solution shows that the magnetic field has a logarithmic spiral component and no definite orientation of the inner contours. Such magnetized vortices may be expected to exist in MHD turbulence with weak magnetic field where the field shows a tendency to align itself with vorticity. Magnetized vortices may also be expected to exist on the solar surface near the corners of convection cells where downwelling swirling flow tends to concentrate the magnetic field.

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter a continuous liquid layer and to apply it to a continuous flexible substrate. Of the two rolls that make a forward or reverse roll coating gap, one is often covered by a layer of more-or-less deformable elastomer. Liquid carried into the converging side of the nip can develop high enough pressure to deform the resilient cover, which changes the nip profile and thus alters the velocity and pressure field. This elastohydrodynamic coupled action is not yet well understood. Theoretical modelling has to take into account the viscous flow, the roll deformation and the free-surface effects in order to predict the flow behaviour.

In this work the flow between a rigid and a deformable counter-rotating roll that shares the same angular speed is described by the lubrication approximation together with a viscocapillary model, based in the film-flow equation, for the film-split region. The deformation of the elastomer layer is modelled by Hookean springs oriented radially, which constitute a one-dimensional model. The stability of the system to transverse perturbation is analysed by examining the time-dependent response to infinitesimal disturbances in order to identify those that grow fastest.

The corresponding system of equations is solved by Newton's method with first-order continuation. The relationship between coating thickness, operational parameters (loading force, gap setting, roll velocities, etc.), liquid properties and the properties of the cover is reported, as well as the critical capillary number for onset of ribbing and wavelengths of the ribbing pattern predicted by the mathematical model. The results indicate how a deformable cover may be used in order to delay the onset of ribbing for a desired coating thickness.

In order to validate the theoretical predictions of the viscocapillary/Hookean spring model, the symmetric film-split flows between a pair of rigid rolls and a pair consisting of a deformable roll and a rigid one were also analysed experimentally.

New laboratory experiments have been performed in a rotating fluid annulus, subject to internal heating and sidewall cooling, in which a radial depth gradient has been created by the inclusion of oppositely sloping boundaries. Endwall configurations that cause the fluid depth (D) to increase with radius (∂D/∂r>0) and to decrease with radius (∂D/∂r<0) have been studied, as the former is applicable to the terrestrial atmosphere and oceans, while the latter may be relevant to deep atmospheres such as those of the giant planets.

Even with the steepest boundary slopes, isolated or periodic chains of stable coherent eddies are observed with both endwall configurations, and these regular eddy modes are seen to drift relative to the walls of the convection chamber concordant with simple Rossby wave ideas. When the boundary slope (δ) is small, no difference is observed in the range of azimuthal wavenumbers seen in the regular wave regimes of the two endwall configurations. At larger values of δ, however, this symmetry is lost, since regular modes m=2 to 8 are observed with ∂D/∂r>0 endwalls, while only a large vertically trapped anticyclonic gyre is seen with ∂D/∂r<0 endwalls. The other effects of the radial depth gradient are the observed reduction in both the lateral and vertical scale of the eddy features, and the formation of two independent trains of eddies within the gap width at sufficiently high rotation rates in the ∂D/∂r>0 endwall experiments. The zonal mean flow is also found to develop a significant barotropic component, superimposed on the vertically and horizontally sheared zonal jets generated by the non-monotonic thermal gradient of the experiment. This barotropic component is predominantly prograde (retrograde) in the ∂D/∂r>0 (∂D/∂r<0) endwall experiments, and confined close to the outer (inner) wall where the fluid depth is greatest.

There is evidence of the formation of increased numbers of zonal jets in the ∂D/∂r>0 endwall experiments above that expected from the form of the thermal forcing. These multiple zonal jets are highly localized in the vertical, and are trapped close to the top boundary. Their radial scale is, nevertheless, close to that given by the Rhines argument. No comparable increase in the radial wavenumber of the mean flow is observed in the ∂D/∂r<0 endwall experiments in the present system.

In this paper we present the results from numerical calculations, based upon the Navier–Stokes equations at relatively high Reynolds number, of the formation of a vortex ring when fluid is ejected from a circular tube. Our results are compared with the experiments of Didden (1979), and the inviscid flow calculations of Nitsche & Krasny (1994). Reasonable agreement is achieved except for the rate of shedding of circulation during the initial stages of ring formation. The theoretically predicted rate of shedding is substantially higher than that predicted by Didden. By contrast the inviscid theory predicts an anomalously high rate of initial shedding. We offer explanations for both of these apparent discrepancies.

Viscous flow computations are required to predict the heat flux or the viscous drag on an hypersonic re-entry vehicle. When real gas effects are included, Navier–Stokes computations are very expensive, whereas the use of standard boundary layer approximations does not correctly account for the ‘entropy layer swallowing’ phenomenon. The purpose of this paper is to present an extension of a new boundary layer theory, called the ‘defect approach’, to two-dimensional hypersonic flows including chemical and vibrational non-equilibrium phenomena. This method ensures a smooth matching of the boundary layer with the inviscid solution in hypersonic flows with strong entropy gradients. A new set of first-order boundary layer equations has been derived, using a defect formulation in the viscous region together with a matched asymptotic expansions technique. These equations and the associated transport coefficient models as well as thermochemical models have been implemented. The prediction of the flow field around the blunt-cone wind tunnel model ELECTRE with non-equilibrium free-stream conditions has been done by solving first the inviscid flow equations and then the first-order defect boundary layer equations. The numerical simulations of the boundary layer flow were performed with catalytic and non-catalytic conditions for the chemistry and the vibrational mode. The comparison with Navier–Stokes computations shows good agreement. The wall heat flux predictions are compared to experimental measurements carried out during the MSTP campaign in the ONERA F4 wind tunnel facility. The defect approach improves the skin friction prediction in comparison with a classical boundary layer computation.

We investigate the global linear stability properties of the quasi-parallel flow in the neighbourhood of the trailing edge of a thin aerofoil, using a WKBJ/multiple scales formulation in the limit of large Reynolds number, as originally developed by Monkewitz, Huerre & Chomaz. We show that the wake is globally linearly unstable to second order in the asymptotic expansion parameter at all Reynolds numbers provided the effective adverse pressure gradient at the trailing edge, which is related to the aerofoil thickness distribution, is sufficiently large. For smaller adverse pressure gradients, there exists a critical Reynolds number above which the flow is globally linearly unstable, but below which it is globally stable. An asymptotic analysis for large wavenumber indicates that the double Blasius profile, corresponding to a zero adverse pressure gradient, may be absolutely unstable.

This paper investigates the effects of bubble dynamics on the stability of parallel bubbly flows of low void fraction. The equations of motion for the bubbly mixture are linearized for small perturbations and the parallel flow assumption is used to obtain a modified Rayleigh equation governing the inviscid stability problem. This is then used for the stability analysis of two-dimensional shear layers, jets and wakes. Inertial effects associated with the bubble response and energy dissipation due to the viscosity of the liquid, the heat transfer between the two phases, and the liquid compressibility are included. Numerical solutions of the eigenvalue problems for the modified Rayleigh equation are obtained by means of a multiple shooting method. Depending on the characteristic velocities of the various flows, the void fraction, and the ambient pressure, the presence of air bubbles can induce significant departures from the classical stability results for a single-phase fluid.

In this study, the interaction with a free surface of an initially axisymmetric jet issuing beneath and parallel to the surface was examined. The purpose was to determine the origin of the ‘surface current’ – the large outward velocity which exists in a thin layer adjacent to the surface. Using the equations of mean motion, it is shown that near the surface, outward acceleration results from the balance between a positive contribution from the lateral Reynolds-stress gradients and a negative contribution from the lateral pressure gradient. The local pressure field near the free surface is shown to be largely determined by the local Reynolds-stress field. Combining these results shows that the lateral acceleration which results in the surface current is related to the Reynolds-stress anisotropy near the surface. The results indicate that there should be roughly a three-fold increase in the lateral growth rate of the jet near the free surface and a similar increase in the outward velocity, when compared to a deep jet. Comparison to available experimental data showed that the maximum outward velocity was consistent with the theory, and that the lateral scale of the surface-current layer was roughly double that of the deep jet, slightly smaller than expected. The near-surface stress anisotropy was shown to be related to the interaction of vorticity with the free surface. This indicates that the results of this study are consistent with earlier explanations of the surface current in terms of vortex/free-surface interaction.

Non-intrusive measurements of the streamwise velocity in turbulent round jets in air are performed by recording short-time displacements and distorsions of very thin tagging lines written spanwise into the flow. The lines are written by Raman-exciting oxygen molecules and are interrogated by laser-induced electronic fluorescence (relief). This gives access to the spatial structure of transverse velocity increments without recourse to the Taylor hypothesis. The resolution is around 25 μm, less than twice the Kolmogorov scale η for the experiments performed (with Rλ≈360–600).

The technique is validated by comparison with results obtained from other techniques for longitudinal or transverse structure functions up to order 8. The agreement is consistent with the estimated errors – a few percent on exponents determined by extended-self-similarity – and indicates significant departures from Kolmogorov (1941) scaling.

Probability distribution functions of transverse velocity increments Δu over separations down to 1:8η are reported for the first time. Violent events, with Δu comparable to the r.m.s. turbulent velocity fluctuation, are found to take place with statistically significant probabilities. The shapes of the corresponding lines suggest the effect of intense slender vortex filaments.

Water wave diffraction by an array of bottom-mounted circular cylinders is analysed under the assumptions of linear theory. The cylinders are identical, and equally spaced along the array. When the number of cylinders is large, but finite, near-resonant modes occur between adjacent cylinders at critical wavenumbers, and cause unusually large loads on each element of the array. These modes are associated with the existence of homogeneous solutions for the diffraction by an array which extends to infinity in both directions. This phenomenon is related to the existence of trapped waves in a channel. A second trapped wave is established, corresponding to Dirichlet boundary conditions on the channel walls, as well as a sequence of higher wavenumbers where ‘nearly trapped’ modes exist.

The trapped modes which can occur in a long narrow wave channel containing any number of different-sized bottom-mounted circular cylinders arbitrarily spaced along the centreline of the channel are considered. The modes, all of which are antisymmetric with respect to the centreplane of the channel are of two types: Neumann modes, in which the fluid has normal velocity zero on the channel walls corresponding to a localized sloshing near the cylinders, and Dirichlet modes, in which the dynamic pressure vanishes on the channel walls. These latter modes have no physical meaning in the water-wave context but have been observed in a related acoustic context where the same governing equations and boundary conditions apply.

It is shown that in general there are [les ]N trapped modes for any configuration of N cylinders, the precise number depending critically on the geometry of the configuration. Both types are of importance in predicting the exciting forces on individual cylinders within a large but finite periodic arrangement of cylinders.

Six subgrid models for the turbulent stress tensor are tested by conducting large-eddy simulations (LES) of the weakly compressible temporal mixing layer: the Smagorinsky, similarity, gradient, dynamic eddy-viscosity, dynamic mixed and dynamic Clark models. The last three models are variations of the first three models using the dynamic approach. Two sets of simulations are performed in order to assess the quality of the six models. The LES results corresponding to the first set are compared with filtered results obtained from a direct numerical simulation (DNS). It appears that the dynamic models lead to more accurate results than the non-dynamic models tested. An adequate mechanism to dissipate energy from resolved to subgrid scales is essential. The dynamic models have this property, but the Smagorinsky model is too dissipative during transition, whereas the similarity and gradient models are not sufficiently dissipative for the smallest resolved scales. In this set of simulations, at moderate Reynolds number, the dynamic mixed and Clark models are found to be slightly more accurate than the dynamic eddy-viscosity model. The second set of LES concerns the mixing layer at a considerably higher Reynolds number and in a larger computational domain. An accurate DNS for this mixing layer can currently not be performed, thus in this case the LES are tested by investigating whether they resemble a self-similar turbulent flow. It is found that the dynamic models generate better results than the non-dynamic models. The closest approximation to a self-similar state was obtained using the dynamic eddy-viscosity model.

A radial temperature distribution is applied to the top of a cylinder of rotating stably stratified fluid. Thermal wind shear drives the interior flow. Linearized theory predicts, and laboratory experiments confirm, that when the stratification is large enough it completely suppresses the Ekman pumping into the interior. The interior velocity field, which is primarily azimuthal, responds by satisfying the no-slip boundary conditions without the need of Ekman layers on the horizontal surfaces. Moreover, for large stratification a thermal boundary layer beneath the top surface traps the thermal disturbance applied at the upper surface. The greatest azimuthal velocity occurs at the base of this layer. Below this layer the azimuthal velocity viscously diffuses downward with thermal wind adjusting the temperature. The Rossby radius of deformation based on this layer depth is the cylinder's radius divided by the square root of the Prandtl number. Detailed measurements of the velocity field generated in the cylinder by the heating are compared with the theory in the case where the Ekman layers are eliminated by stratification. The theory and experiments agree qualitatively well over a range of four orders of magnitude of imposed parameters and over a large parameter range the quantitative comparison is also very good.